Failing Linear Algebra:
flip
news:c98b1ba0.04042...@posting.google.com...
> I'm currently a math major and am taking linear algebra, but I'm in
> serious danger of failing. I just don't get it! Is this newsgroup a
> place to come to ask questions and get information about learning
> math? Or is there somewhere more appropriate to go? I've always had
> trouble with vectors, and I think I fell apart sort of right at the
> beginning of linear algebra (although, I did manage to get a B- on the
> very first exam). I've got another exam next week. What can I do? I
> don't get all the terms, concepts, and jargon. Anyone know how to
> make learning linear algebra easier and more practical? Anyone got
> any practice problems?
1. "3000 Solved Problems in Linear Algebra", Schaum's Outline, by Seymour
Lipschutz (want to practice problems ... here you go!)
2. "Elementary Linear Algebra, APplications Version", by H. Anton and Chris
Rorres
3. "Applied Linear Algebra" by Ben Noble and james W. Daniel
4. "Linear Algebra, An Introduction with Concurrent Examples", by A. G.
Hamilton
HTH, Flip
Michael N. Christoff
>
> 2. "Elementary Linear Algebra, APplications Version", by H. Anton and
Chris
> Rorres
>
I certainly agree with this recomendation. While not the most exhaustive
book on linear algebra, its explanations and proofs of what it does have are
some of the clearest I've seen. Also, it gets right to the point with no
unneccesary commentary or musings. Perfect if you're short on time.
l8r, Mike N. Christoff
Gerry Myerson
de...@aol.com (DE781) wrote:
> I'm currently a math major and am taking linear algebra, but I'm in
> serious danger of failing. I just don't get it! Is this newsgroup a
> place to come to ask questions and get information about learning
> math? Or is there somewhere more appropriate to go?
The appropriate place to go is your teacher's office. Ask her to
take the time to go through a few problems with you & to suggest
what you ought to be doing to get a grip on the material.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
Dave Rusin
DE781 <de...@aol.com> wrote:
>I'm currently a math major and am taking linear algebra, but I'm in
>serious danger of failing. I just don't get it! Is this newsgroup a
>place to come to ask questions and get information about learning
>trouble with vectors, and I think I fell apart sort of right at the
>beginning of linear algebra (although, I did manage to get a B- on the
>very first exam). I've got another exam next week. What can I do? I
>don't get all the terms, concepts, and jargon. Anyone know how to
>make learning linear algebra easier and more practical? Anyone got
>any practice problems?
I would bet that the single practice problem you need to work on is,
"What is a vector space?"
Our students "do well on the very first exam" because that's the part
of the course where we warm up with techniques for solving linear
systems of equations and such topics. But our LA course is also our
students' first course in which abstractions, axioms, and proofs play
a significant role. They often stumble because (among other problems)
they don't realize they need to _memorize_ definitions _precisely_.
So you can do a little self-assessment here to figure out whether
what you're missing is bits of topics or the core idea: can you,
right this minute, define what a vector space is?
dave
Acid Pooh
> I'm currently a math major and am taking linear algebra, but I'm in
> serious danger of failing. I just don't get it! Is this newsgroup a
> place to come to ask questions and get information about learning
> math? Or is there somewhere more appropriate to go? I've always had
> trouble with vectors, and I think I fell apart sort of right at the
> beginning of linear algebra (although, I did manage to get a B- on the
> very first exam). I've got another exam next week. What can I do? I
> don't get all the terms, concepts, and jargon. Anyone know how to
> make learning linear algebra easier and more practical? Anyone got
> any practice problems?
I'd like to help you, but some specifics would be necessary. What
terminology are you having trouble with?
At any rate, this is the perfect opportunity to see if you have what
it takes to be a math major (or mathematician, if that's your goal).
You need to learn a method to go over the material quickly and
thoroughly. You need to learn how to come up with intuitive
representations of the concepts involved to guide your thinking. You
need to learn how to apply what you know to problem solving.
So, what I suggest is that you stick it out. Read the definitions
until your head hurts. Try to figure out exactly what the quantifiers
are telling you. Do problems until your head hurts more and you dream
about mathematical symbols. Don't let a problem intimidate you.
If this sounds appealling, you probably have what it takes. If not, I
would suggest another major--it's not going to get any easier after
linear algebra.
'cid 'ooh
(I'm not trying to scare you. I'm a student too, and I'm really
excited about facing new challenges in grad school. But I often have
to work for 48 hours straight to figure out a few measly problems.
Realistic expectations are important in any endeavor of this sort)
Thomas Nordhaus
>a significant role. They often stumble because (among other problems)
>they don't realize they need to _memorize_ definitions _precisely_.
Memorize? I can't remember ever memorizing anything. Better just to
practice until you understand. Discuss, ask questions, apply. That way
you memorize, of course, but that's just a side-effect.
>So you can do a little self-assessment here to figure out whether
>what you're missing is bits of topics or the core idea: can you,
>right this minute, define what a vector space is?
Hopefully not just as a one-to-one rendition of phrases from a
textbook...
Thomas
>
>dave
Chan-Ho Suh
Nordhaus <thnor...@yahoo.de> wrote:
> ru...@vesuvius.math.niu.edu (Dave Rusin) schrieb:
>
> >a significant role. They often stumble because (among other problems)
> >they don't realize they need to _memorize_ definitions _precisely_.
>
> Memorize? I can't remember ever memorizing anything. Better just to
> practice until you understand. Discuss, ask questions, apply. That way
> you memorize, of course, but that's just a side-effect.
>
It's nice to think that all the students are going to go home and
follow your advice and "discuss, ask questions, apply", but they're
not. Most will not, even the better ones.
Failing a class sucks. Dave's advice to memorize the definitions
precisely is good, because a large part of the points for a beginning
linear algebra course is based on reciting the definitions. The way
the typical service course is rigged is to weed out the people who
can't memorize or do the rote computation.
We're talking about someone who probably doesn't even remember enough
to discuss anything about vector spaces. You may not have ever
memorized a definition (seems remarkable, but possible), but you have
undoubtedly partially memorized them enough to discuss them, e.g. "Ah,
the definition of a vector space, doesn't it have something to do with
scalar multiplication and an abelian group..."
Of course, when confronted with a definition, one should try and
understand it. Unfortunately, in this context, the reality is that few
students can even vaguely remember what it is they read, and they don't
go back and reread it until they have to (when they're flunking).
> >So you can do a little self-assessment here to figure out whether
> >what you're missing is bits of topics or the core idea: can you,
> >right this minute, define what a vector space is?
>
> Hopefully not just as a one-to-one rendition of phrases from a
> textbook...
>
Actually, I know too much to even hope for this from the average
student.
David C. Ullrich
<s...@math.ucdavis.nospam.edu> wrote:
>In article <obdh809dt6ktsqme0...@4ax.com>, Thomas
>Nordhaus <thnor...@yahoo.de> wrote:
>
>> ru...@vesuvius.math.niu.edu (Dave Rusin) schrieb:
>>
>> >a significant role. They often stumble because (among other problems)
>> >they don't realize they need to _memorize_ definitions _precisely_.
>>
>> Memorize? I can't remember ever memorizing anything. Better just to
>> practice until you understand. Discuss, ask questions, apply. That way
>> you memorize, of course, but that's just a side-effect.
>>
>
>It's nice to think that all the students are going to go home and
>follow your advice and "discuss, ask questions, apply", but they're
>not. Most will not, even the better ones.
>
>Failing a class sucks. Dave's advice to memorize the definitions
>precisely is good, because a large part of the points for a beginning
>linear algebra course is based on reciting the definitions.
It's excellent advice for what seems to me to be a much more
important reason:
Of course actually _understanding_ the definition would be much
better than just memorizing it verbatim. But, as I tell my students
over and over in this sort of class, it's exactly the things you don't
really understand at first that you _need_ to memorize _precisely_.
If one knows a bunch of meaningless definitions, precisely,
then at some later point it may happen that one sees how
they fit together - seeing how the definitions fit together _is_
understanding them, as far as I'm concerned.
Otoh if one has a fuzzy version of the definitions in mind
then one _cannot_ see later what sense it all makes,
because the stuff one has in mind _doesn't_ make
precise sense!
>The way
>the typical service course is rigged is to weed out the people who
>can't memorize or do the rote computation.
>
>We're talking about someone who probably doesn't even remember enough
>to discuss anything about vector spaces. You may not have ever
>memorized a definition (seems remarkable, but possible), but you have
>undoubtedly partially memorized them enough to discuss them, e.g. "Ah,
>the definition of a vector space, doesn't it have something to do with
>scalar multiplication and an abelian group..."
>
>Of course, when confronted with a definition, one should try and
>understand it. Unfortunately, in this context, the reality is that few
>students can even vaguely remember what it is they read, and they don't
>go back and reread it until they have to (when they're flunking).
>
>
>> >So you can do a little self-assessment here to figure out whether
>> >what you're missing is bits of topics or the core idea: can you,
>> >right this minute, define what a vector space is?
>>
>> Hopefully not just as a one-to-one rendition of phrases from a
>> textbook...
>>
>
>Actually, I know too much to even hope for this from the average
>student.
************************
David C. Ullrich
David C. Ullrich
>I'm currently a math major and am taking linear algebra, but I'm in
>serious danger of failing. I just don't get it! Is this newsgroup a
>place to come to ask questions and get information about learning
>math? Or is there somewhere more appropriate to go? I've always had
>trouble with vectors, and I think I fell apart sort of right at the
>beginning of linear algebra (although, I did manage to get a B- on the
>very first exam). I've got another exam next week. What can I do? I
>don't get all the terms, concepts, and jargon. Anyone know how to
>make learning linear algebra easier and more practical? Anyone got
>any practice problems?
My answer to this is a little long. I hope you'll read all of it: I've
taught that course probably 20 times in the last 29 years;
I know two things from experience: (i) what I have to say is
good advice (ii) you're not going to believe it's good advice.
The long part of this post is an attempt to explain why it's
good advice (also some actual empircal evidence that it
_is_ good advice, whether you believe it or not).
The advice is what Rusin already said: next time you take the
course _learn_ the definitions, _precisely_. Word for word.
You need to know the definitions well enough that when you
see the word "basis" the phrase "independent spanning set"
pops into your head _immediately_, without a moment's
thought.
Probably it's too late for that this semester, because there's
a _large_ number of definitions you need to learn. You
need to learn each one, _precisely_, as soon as it comes
up in class.
Why:
I teach that class a lot. A lot of students have a lot of trouble.
I always give the class the advice above. Almost all the
students simply ignore the advice - if they do bother to
try to actually learn the definitions it's just the day before
the test. I think this is because "memorize" is sort of a
dirty word - our attitude these days is that we're not
supposed to be memorizing things, we're suppposed to
be "understanding" them. I tell me students they need
to learn the definitions and they say "you mean _memorize_
them?", as though they can't believe I'd suggest such
a thing.
One way to look at it is this: you're trying to learn a
new _language_. When you take German you don't
find anything strange about the fact that you have to
simply memorize what the words mean - _after_ you
memorize what the words mean you can understand
sentences written in German; trying to understand
German prose _before_ simply learning what the
words mean is simply ridiculous. But students try
to do the equivalent thing in linear algebra.
The thing is you're dealing with an _abstract_
subject. If you're studying potatoes you don't
need a definition of "potato", because you already
know what a potato is. But if you're studying
vector spaces, linear operators, etc, you do need
to know the definitions, because a vector space
is not something you can point to, a vector space
is exactly what the definition says it is.
Of course _understanding_ the definitions is the
actual goal. But in an _abstract_ subject what it
_means_ to understand a definition is to see
exactly how it fits in with the _other_ definitions.
If you know all the definitions, _precisely_, then
it may happen at some point you _will_ see how
they all fit together, and at that point voila, you
have understood the definitions! But if you don't
know _exactly_ what the definitions say then it
_can't_ happen that at some point you will see
exactly how they all fit together, because your
fuzzy versions of the definitions will _not_ fit
together the way they're supposed to!
So much for the abstract explanation. Now for
the empirical evidence:
I teach that course a lot. I always ask for a lot
of definitions on quizzes and tests. Not because
I think that being able to recite the definitions should
actually be the goal, but because I know that knowing
the definitions is essential to being able to work the
problems, and I also know that the students don't
believe that. (What really puzzles me is that they
don't find the fact that I'm going to be asking for
the defintions on quizzes and tests to be enough
motivation to learn them - it happens that I ask
for the definiton on "basis" four weeks in a row,
all they have to say is "independent spanning
set", and some students will get the very same
question wrong four weeks in a row. Whatever...)
Now, not all the people here who teach this class
do ask for definitions on quizzes and tests. Here's
the good part: It's happened several times that
someone takes the course from me and flunks.
He retakes the course from someone else.
Even though the other professor _doesn't_
ask for definitions on quizzes and tests he
decides to give it a try. I run into the student
later, and he tells me that he tried "my" way,
and sure enough it worked! He re-took the
course, _learned_ all the definitions rock-solid,
and he got an A or a B in the course!
NOTE that he didn't get that A or B the second
time because he got the definitions right on
tests. The other professor wasn't asking for
definitions. He got the A or B because sure enough,
it turned out that knowing the definitions allowed
him to actually do the problems.
Try it next time. (Or don't, but it's the only way,
honest).
************************
David C. Ullrich
John T Lowry
failing" post. I was surprised that no one emphasized two aspects of
learning linear algebra which, to me, seem crucial:
1. getting a clear geometric idea of bases, subspaces, kernels, etc., and
2. getting a clear idea, mostly through non-trivial examples, of the
difference between such abstract entities as vectors and operators as
against their concrete coordinatizations (row, column, and rectangular
matrices). While it's probably NOT necessary, in this first course, to get
the covariant/contravariant tangle straightened out, I think it's certainly
necessary to be clear on how to represent linear operators using alternative
bases in both the domain and range spaces. Kemeny, Mirkl, Snell, and
Thompson's fairly elementary book (I once taught a good portion of it to
high school students at Colorado Rocky Mountain School) Finite Mathematical
Structures, is good on that last item.
HTH
John
--
John T Lowry, PhD
Flight Physics
5217 Old Spicewood Springs Rd, #312
Austin, Texas 78731
(512) 231-9391
jlow...@earthlink.net
"DE781" <de...@aol.com> wrote in message
news:c98b1ba0.04042...@posting.google.com...
Jonathan Miller
news:gerry-1AD269....@sunb.ocs.mq.edu.au...
> In article <c98b1ba0.04042...@posting.google.com>,
> de...@aol.com (DE781) wrote:
>
> > I'm currently a math major and am taking linear algebra, but I'm in
> > serious danger of failing. I just don't get it! Is this newsgroup a
> > place to come to ask questions and get information about learning
> > math? Or is there somewhere more appropriate to go?
>
> The appropriate place to go is your teacher's office. Ask her to
> take the time to go through a few problems with you & to suggest
> what you ought to be doing to get a grip on the material.
Seriously. She gets paid for that. Not to teach you (you have to learn for
yourself), but to help you learn.
And, of course, to publish.
Jon Miller
Daniel Grubb
>>a significant role. They often stumble because (among other problems)
>>they don't realize they need to _memorize_ definitions _precisely_.
>Memorize? I can't remember ever memorizing anything. Better just to
>practice until you understand. Discuss, ask questions, apply. That way
>you memorize, of course, but that's just a side-effect.
No, for proof classes, which linear algebra is in many places,
it is crucial to *memorize* the definitions. This holds for
all the proof classes at higher levels also. There is simply no
way of giving rigorous proofs if you don't know the actual
definitions. All too often, students have some very vague ideas
of what is going on and then can't even get started on a proof
because they don't know the *exact* definiton used in the course.
>>So you can do a little self-assessment here to figure out whether
>>what you're missing is bits of topics or the core idea: can you,
>>right this minute, define what a vector space is?
>Hopefully not just as a one-to-one rendition of phrases from a
>textbook...
The student should be able to give a rendition that is at least
equivalent to the one given in the book and that uses precise language.
If you can't say that a basis is an independent spanning set
then you don't know what a basis is. If you can't give the quantifiers
for the definition of independence, you won't be able to do a
proof using independence.
--Dan Grubb
James Dolan
daniel grubb <gr...@math.niu.edu> wrote:
|>>a significant role. They often stumble because (among other problems)
|>>they don't realize they need to _memorize_ definitions _precisely_.
|
|>Memorize? I can't remember ever memorizing anything. Better just to
|>practice until you understand. Discuss, ask questions, apply. That way
|>you memorize, of course, but that's just a side-effect.
|
|No, for proof classes, which linear algebra is in many places,
|it is crucial to *memorize* the definitions.
nonsense.
|This holds for
|all the proof classes at higher levels also. There is simply no
|way of giving rigorous proofs if you don't know the actual
|definitions.
it tends to be crucial to know the actual definitions at the moment
that you're giving a rigorous proof. since at least a few thousand
years ago this can be accomplished simply by having some technical
device (book, computer, crib sheet, or the like) do the memorization
for you. it's hardly necessary to do anything as totally insane as
wasting human effort on rote memorization.
--
[e-mail address jdo...@math.ucr.edu]
Marc Olschok
>
>>>a significant role. They often stumble because (among other problems)
>>>they don't realize they need to _memorize_ definitions _precisely_.
>
>>Memorize? I can't remember ever memorizing anything. Better just to
>>practice until you understand. Discuss, ask questions, apply. That way
>>you memorize, of course, but that's just a side-effect.
>
> No, for proof classes, which linear algebra is in many places,
> it is crucial to *memorize* the definitions. This holds for
> all the proof classes at higher levels also. There is simply no
> way of giving rigorous proofs if you don't know the actual
> definitions. All too often, students have some very vague ideas
> of what is going on and then can't even get started on a proof
> because they don't know the *exact* definiton used in the course.
No disagreement here. Perhaps Thomas only meant to warn against
blind memorization without understanding. This way a student would
end up with the ability to repeat the memorized definition but
nothing else. This danger is even greater, when those definitions
are memorized as isolated entities, without checking them against
examplex, counterexamples and proofs.
Marc
rus...@mdli.com
<ull...@math.okstate.edu> wrote:
[snip]
>The advice is what Rusin already said: next time you take the
>course _learn_ the definitions, _precisely_. Word for word.
>You need to know the definitions well enough that when you
>see the word "basis" the phrase "independent spanning set"
>pops into your head _immediately_, without a moment's
>thought.
>
>Probably it's too late for that this semester,
Although this professor's pessimism is warranted, based on his
experience and on what little we know of the OP from the posting, that
doesn't necessarily mean the *student* should be pessimistic yet.
There's a whole weekend (plus?) before the test, and the concepts
*can* be learned in that amount of time if sufficient organization and
diligence are put in. Talk to your professor if possible, and in any
case plan out your study so you can follow Rusin's excellent advice
first and still have time to do some problems and proofs (which may
begin to make more sense to you). Consider it a challenge.
Of course you should also not be reading Usenet any more this week, so
please log off now.
[snip good stuff]
Brian Borchers
Ullrich's advice is correct as far as it goes. I have a few
additional suggestions:
Suppose that you have a definition like:
A <frotz> is a <mumble> that has the properties
1. <foo>
2. <bar>
3. <plugh>
and theorems like:
In any <frotz>, <x> happens.
You'll find that lots of exercises are of the form:
Consider <plover>. Is <plover> a <frotz>?
When you solve such a problem it may be that your grader will accept a
"yes" or "no" answer, and it's likely that the answer in the back of
the book will be a simple "yes" or "no". Do not be tempted by this.
If the answer is "yes", then your answer should be of the form "Yes,
because <plover> has properties <foo>, <bar>, and <plugh>", followed
by work that shows this. On the other hand, if the answer is "no",
then your answer should be something like, "No, because <plover> does not
satisfy <bar>", followed by work that shows this.
The important point here is that mathematics is not simply a guessing game.
It's really about reading and understanding logical arguments, and then later
constructing your own logical arguments.
You should also take time after memorizing the definition and the
theorem to come up with examples and counterexamples related to this
definition. Start by coming up with a <frotz>, verifying that is has
properties <foo>, <bar> and <plugh>, and that <x> happens. Then
construct something that is almost a <frotz> but doesn't satisfy
property <foo>. Does <x> happen? If it doesn't, then you can see one
reason why the property <foo> is part of the definition of a <frotz>.
In general, you need to explore each of the parts of the definition, and
understand what "goes wrong" when one of the required properties is not
satisfied.
--
Brian Borchers borc...@nmt.edu
Department of Mathematics http://www.nmt.edu/~borchers/
New Mexico Tech Phone: 505-835-5813
Socorro, NM 87801 FAX: 505-835-5366
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Thomas Nordhaus
>In article <obdh809dt6ktsqme0...@4ax.com>, Thomas
>Nordhaus <thnor...@yahoo.de> wrote:
>
>> ru...@vesuvius.math.niu.edu (Dave Rusin) schrieb:
>>
>> >a significant role. They often stumble because (among other problems)
>> >they don't realize they need to _memorize_ definitions _precisely_.
>>
>> Memorize? I can't remember ever memorizing anything. Better just to
>> practice until you understand. Discuss, ask questions, apply. That way
>> you memorize, of course, but that's just a side-effect.
>>
>
>It's nice to think that all the students are going to go home and
>follow your advice and "discuss, ask questions, apply", but they're
>not. Most will not, even the better ones.
>
>Failing a class sucks. Dave's advice to memorize the definitions
>precisely is good, because a large part of the points for a beginning
>linear algebra course is based on reciting the definitions. The way
>the typical service course is rigged is to weed out the people who
>can't memorize or do the rote computation.
OK. My comment came from my own experience during my study (70s,
Germany, 80s USA). The situation then / there was probably very much
different from the situation you're talking about. The time was
anti-authoritative. We were protesting and boycotting all the time (or
so I seem to remember). Professors wore long hair and were members of
exotic communist parties. It was *very* liberal. We hadn't any exams
(had to do assignments of course). Classes were small...
Later at graduate school in USA, work was very hard. You could barely
finish the assignments, had to teach Calculus-courses and such things.
But you could simply not do it by way of memorizing. You just had to
solve problems and apply theory all the time. So memorizing came just
along the way.
Fact is, that I (fortunately) never *had* to study by way of
memorizing, "cramming".
>
>We're talking about someone who probably doesn't even remember enough
>to discuss anything about vector spaces. You may not have ever
>memorized a definition (seems remarkable, but possible),
As I said, you'll memorize as a side-effect.
Thomas
Daniel Grubb
>|>Memorize? I can't remember ever memorizing anything. Better just to
>|>practice until you understand. Discuss, ask questions, apply. That way
>|>you memorize, of course, but that's just a side-effect.
>|
>|No, for proof classes, which linear algebra is in many places,
>|it is crucial to *memorize* the definitions.
>nonsense.
I'm afraid not. It really is crucial to know the definitions
once you get past the lower level undergraduate courses in math.
>|This holds for
>|all the proof classes at higher levels also. There is simply no
>|way of giving rigorous proofs if you don't know the actual
>|definitions.
>it tends to be crucial to know the actual definitions at the moment
>that you're giving a rigorous proof. since at least a few thousand
>years ago this can be accomplished simply by having some technical
>device (book, computer, crib sheet, or the like) do the memorization
>for you. it's hardly necessary to do anything as totally insane as
>wasting human effort on rote memorization.
Um. All math classes after the elementary level are *proof* classes.
The students better know those definitons inside and out so they can
do the needed proofs on exams, qualifiers, etc.
--Dan Grubb
Daniel Grubb
>> No, for proof classes, which linear algebra is in many places,
>> it is crucial to *memorize* the definitions. This holds for
>> all the proof classes at higher levels also. There is simply no
>> way of giving rigorous proofs if you don't know the actual
>> definitions. All too often, students have some very vague ideas
>> of what is going on and then can't even get started on a proof
>> because they don't know the *exact* definiton used in the course.
>No disagreement here. Perhaps Thomas only meant to warn against
>blind memorization without understanding. This way a student would
>end up with the ability to repeat the memorized definition but
>nothing else. This danger is even greater, when those definitions
>are memorized as isolated entities, without checking them against
>examplex, counterexamples and proofs.
I agree. After you have those definitions memorized, you have to
get understanding of those definitions through theorems and examples,
which show unforseen consequences of those definitions.
--Dan Grubb
James Dolan
daniel grubb <gr...@math.niu.edu> wrote:
|>|>Memorize? I can't remember ever memorizing anything. Better just to
|>|>practice until you understand. Discuss, ask questions, apply. That way
|>|>you memorize, of course, but that's just a side-effect.
|>|
|>|No, for proof classes, which linear algebra is in many places,
|>|it is crucial to *memorize* the definitions.
|
|>nonsense.
|
|I'm afraid not.
in that case you ought to have at least tried to give some reason to
support your opinion.
--
[e-mail address jdo...@math.ucr.edu]
Dave Rusin
>
>>a significant role. They often stumble because (among other problems)
>>they don't realize they need to _memorize_ definitions _precisely_.
>
>Memorize? I can't remember ever memorizing anything. Better just to
>practice until you understand.
Sure. (I've never "memorized" a piece at the piano either. As you
suggest, "memorizing" comes from constant practice.) But some students
believe that "understanding" is equivalent to "exposure", or something.
Here is an example. A student I knew contacted me because she was having
trouble in her Linear Algebra class. The teacher was terrible, etc. etc.
In the course of the conversation I got her to discuss a typical recent
homework exercise and how she had approached it and how the teacher had
responded to her attempt.
The question: "Show that similarity of matrices is an equivalence relation".
In her defense, I will concede that this typical mathematician-speak is
just a bit cryptic to the neophyte: similarity is really a _concept_, and
not strictly speaking a collection of ordered pairs, as an equivalence
relation is supposed to be. So I decided to help her a bit and asked her
specifically: is every matrix similar to itself? (Symmetry and transitivity
were to follow later.)
There was a pause, and she eventually said something like, "Well, sure:
every matrix is EQUAL to itself. How much more similar can you get?"
Clearly the student had not memorized the definition of similarity; or
maybe she didn't yet understand that in order to prove a statement
"There exists a Q such that..."
we (usually) begin by producing a Q.
It was with this kind of example in mind that I advised the OP to
memorize the definition: a literal parroting of the words is neither
necessary nor sufficient but an accurate rewording is crucial.
dave
Thomas Nordhaus
>
>>>a significant role. They often stumble because (among other problems)
>>>they don't realize they need to _memorize_ definitions _precisely_.
>
>>Memorize? I can't remember ever memorizing anything. Better just to
>>practice until you understand. Discuss, ask questions, apply. That way
>>you memorize, of course, but that's just a side-effect.
>
>No, for proof classes, which linear algebra is in many places,
>it is crucial to *memorize* the definitions. This holds for
>all the proof classes at higher levels also.
I disagree. As I said, you memorize (of course). But not by way of
"cramming, swotting" (or "büffeln" - German). Worked for me.
>There is simply no
>way of giving rigorous proofs if you don't know the actual
>definitions.
Absolutely correct. But that doesn't disprove my assertion.
> All too often, students have some very vague ideas
>of what is going on and then can't even get started on a proof
>because they don't know the *exact* definiton used in the course.
When I'm not sure, I go back to my book and look it up (Happened
recently, when I wasn't sure what the exact definition of a refinement
is (in the context of paracompact top. spaces)). You forget, of
course, unless you solve problems or apply in a regular way.
>
>>>So you can do a little self-assessment here to figure out whether
>>>what you're missing is bits of topics or the core idea: can you,
>>>right this minute, define what a vector space is?
>
>>Hopefully not just as a one-to-one rendition of phrases from a
>>textbook...
>
>The student should be able to give a rendition that is at least
>equivalent to the one given in the book and that uses precise language.
Absolutely right. But it should go above Parroting, that's what I
meant.
Thomas
maky m.
> serious danger of failing. I just don't get it!
perhaps math is not for you.
> Is this newsgroup a place to come to ask questions and get
> information about learning math?
to some extent, yes.
> Or is there somewhere more appropriate to go? I've always had
> trouble with vectors, and I think I fell apart sort of right at the
> beginning of linear algebra (although, I did manage to get a B- on the
> very first exam).
it may be the case that your B- wasn't the result of your efforts, but
rather the fear your math teacher has of losing his job.
> I've got another exam next week. What can I do?
study and solve problems?
> I don't get all the terms, concepts, and jargon.
do you read your book? does your math teacher show you examples? do
you do homework?
> Anyone know how to make learning linear algebra easier and more
> practical?
linear algebra has plenty of neat applications, but i guess you would
need to learn LINEAR ALGEBRA first.
> Anyone got any practice problems?
doesn't your book already have those?
Wayne Brown
>
> Sure. (I've never "memorized" a piece at the piano either. As you
> suggest, "memorizing" comes from constant practice.) But some students
> believe that "understanding" is equivalent to "exposure", or something.
Something of which I have to remind myself constantly is that
"understanding" isn't always the same as "learning." Sometimes I read
through a proof and understand each step, and how each follows from the
previous step. But the real test is whether I can come back to it later,
after I've forgotten the details, and reproduce it myself. If I can't,
I haven't really *learned* it. I need not only to be able to follow the
reasoning, but to grasp the overall direction of the proof and understand
why *that* particular approach was taken. If I can't understand both
the details and the "big picture" then I'm just wasting my time. All too
often I read through a chapter and "understand" everything in it, but have
only a hazy idea of the concepts a few days later. That's why I often put
a book aside for a while and then come back and re-read the last couple
of chapters I've read, to be certain I'm really ready to go forward.
--
Wayne Brown (HPCC #1104) | "When your tail's in a crack, you improvise
fwb...@bellsouth.net | if you're good enough. Otherwise you give
| your pelt to the trapper."
"e^(i*pi) = -1" -- Euler | -- John Myers Myers, "Silverlock"
Porker899
particular problems are. Some linear algebra courses are purely computational.
Others mix a little bit of proofs in with the vast majority still being
computational. And then there are those that are almost completely theoretical
with some computations almost as an afterthought.
That said, I also liked Howard Anton's books. He has regular and applied
versions for linear algebra. Choose whatever is most suitable for your class.
Daniel Grubb
>>>Memorize? I can't remember ever memorizing anything. Better just to
>>>practice until you understand. Discuss, ask questions, apply. That way
>>>you memorize, of course, but that's just a side-effect.
>>
>>No, for proof classes, which linear algebra is in many places,
>>it is crucial to *memorize* the definitions. This holds for
>>all the proof classes at higher levels also.
>I disagree. As I said, you memorize (of course). But not by way of
>"cramming, swotting" (or "büffeln" - German). Worked for me.
Hmmmm...Memorizing a definition is not the same as, say, memorizing
digits of pi. At the very least, you should have the quantifiers
in the right place and have something logically equivalent to what
is given in the book. The problem comes when a student comes to me
and I ask what the definition of 'independence' is. The student says
something like 'c_1 v_1 +...+c_n v_n=0 and c_1=...c_n=0'. I'm sorry, this
is not the definition, it is not equivalent to the definition and
a student that uses this as the definition will not be able to either
get the proofs done that are assigned nor will they understand the proofs
in the book, nor will they really understand what independence is until
they get it right. I do see there as being a bit of 'cramming' in
getting it right.
I agree with Ullrich. Learning mathematics is like learning a language.
You have to memorize some things before you can even get off
the ground. After a while, you can pick up meanings quicker
and see the subtleties sooner, but memorization at some level remains
crucial for understanding.
>>There is simply no
>>way of giving rigorous proofs if you don't know the actual
>>definitions.
>Absolutely correct. But that doesn't disprove my assertion.
And they can't understand the theorems or the proofs in the book until
they have the correct definitions. The quickest way, at least at first,
for a student to get the correct definitions is to memorize them.
>> All too often, students have some very vague ideas
>>of what is going on and then can't even get started on a proof
>>because they don't know the *exact* definiton used in the course.
>When I'm not sure, I go back to my book and look it up (Happened
>recently, when I wasn't sure what the exact definition of a refinement
>is (in the context of paracompact top. spaces)). You forget, of
>course, unless you solve problems or apply in a regular way.
Yes, we all forget things over time. But if you are taking a course
in topology that covers paracompactness, I would certainly hope
you have the definition of 'refinement' memorized (by whatever
method) come time for the exam. Or even time to read the next theorem.
If you don't, the definition of paracompactness will be very hard to
understand.
>>The student should be able to give a rendition that is at least
>>equivalent to the one given in the book and that uses precise language.
>Absolutely right. But it should go above Parroting, that's what I
>meant.
Of course it should. But I am lucky to get anything close to an
equivalent statement of the definition from my students. Simply
getting them to understand the difference between 'if...then'
and 'and' has been a struggle.
--Dan Grubb
Abraham Buckingham
> I'm currently a math major and am taking linear algebra, but I'm in
> place to come to ask questions and get information about learning
> trouble with vectors, and I think I fell apart sort of right at the
> beginning of linear algebra (although, I did manage to get a B- on the
> don't get all the terms, concepts, and jargon. Anyone know how to
> make learning linear algebra easier and more practical? Anyone got
> any practice problems?
Get yourself a piece of bristle board. Reproduce all your notes on the
bristle board in compact form, include diagrams and pictures from your
notes and textbook. Every definition in the course should be on the
bristle board, and every important theroem in the course should be on
it as well. Use color and alot of diagrams. Once you're done, put this
on your wall next to your bed. Every morning when you get up, and
every night before you go to sleep read everything on this
bristleboard to yourself outloud. Do this every day for the entire
duration of the course, and do this for every course you take. If you
don't understand anything on this bristle board, then talk to you
teacher about it IMMEDIATELY.
This is the best advice I can give you as someone who'd learned most
everything through home study. There is no royal road to mathamatics,
it's repetition and memorization and those who claim "understanding"
will give you what you really need are fundementally different
students, and by the fact you're nearly failing the course I suggest
you ignore that approach and power though the content in this manner.
Although you MAY not understand it, by memorizing it you at least give
yourself the chance to see it come up in another context later on, and
get the required epiphany.
As for a resource on the internet you can find a complete course in
Linear Algebra at http://ocw.mit.edu/OcwWeb/Mathematics/18-06Linear-AlgebraFall2002/CourseHome/index.htm
thanks to MIT, including video lectures from one Proffessor Strang,
problems to solve, and practice exams. I found these lectures to be
extremely useful in increasing my understanding of the content you're
working with. Good Luck!! :)
Thomas Nordhaus
>
>>When I'm not sure, I go back to my book and look it up (Happened
>>recently, when I wasn't sure what the exact definition of a refinement
>>is (in the context of paracompact top. spaces)). You forget, of
>>course, unless you solve problems or apply in a regular way.
>
>Yes, we all forget things over time. But if you are taking a course
>in topology that covers paracompactness, I would certainly hope
>you have the definition of 'refinement' memorized (by whatever
>method) come time for the exam. Or even time to read the next theorem.
>If you don't, the definition of paracompactness will be very hard to
>understand.
Yes, I didn't understand paracompactness, so I had to read the
textbook, looked through the section on partition of unity and so on.
I wouldn't have time in an exam, so I would have missed the points. In
an examination situation the material would have been a lot closer,
timewise, and I could have reconstructed the definition by reflecting
on the problems that I solved before. Like: "What has to be subset of
what?... Ah, of course!"
Ok, this may not be practical sound advice. If the exam is just
designed for the students to dump factual knowledge in contrast to
solving problems you'll have to memorize because of the sheer
quantity. You have to become a "definition robot". Like - (maybe)
medical doctors, who'll have to recite all the bones in the human body
in alphabetical order.
>
>>>The student should be able to give a rendition that is at least
>>>equivalent to the one given in the book and that uses precise language.
>
>>Absolutely right. But it should go above Parroting, that's what I
>>meant.
>
>Of course it should. But I am lucky to get anything close to an
>equivalent statement of the definition from my students. Simply
>getting them to understand the difference between 'if...then'
>and 'and' has been a struggle.
Hmm, maybe the student should have a minimum amount of talent, maybe
it is as simple as that?
Thomas
>
>--Dan Grubb
David Ames
> In article <obdh809dt6ktsqme0...@4ax.com>, Thomas
> Nordhaus <thnor...@yahoo.de> wrote:
>
> We're talking about someone who probably doesn't even remember enough
> to discuss anything about vector spaces. You may not have ever
> memorized a definition (seems remarkable, but possible), but you have
> undoubtedly partially memorized them enough to discuss them, e.g. "Ah,
> the definition of a vector space, doesn't it have something to do with
> scalar multiplication and an abelian group..."
>
Yup. When I was a senior looking for a job about 10a + b years ago, I
heard about a student who washed out of a job interview when asked
"What is a vector space?" Stage fright? Maybe so, maybe not.
David Ames
Dave Rusin
Thomas Nordhaus <thnor...@yahoo.de> wrote:
> gr...@lola.math.niu.edu (Daniel Grubb) schrieb:
>>Of course it should. But I am lucky to get anything close to an
>>equivalent statement of the definition from my students. Simply
>>getting them to understand the difference between 'if...then'
>>and 'and' has been a struggle.
>
>Hmm, maybe the student should have a minimum amount of talent, maybe
>it is as simple as that?
Elitist! We simply won't put up with that kind of talk.
Mitch Harris
>I'm currently a math major and am taking linear algebra, but I'm in
>serious danger of failing. I just don't get it! Is this newsgroup a
>place to come to ask questions and get information about learning
>math?
1) for informal curiosity questions, sci.math is great
2) if you expect the sci.math news group to help you consistently every
week on your homework problems, it could happen but it depends on your
doing a lot of the work first and being very conscientious in your
requests for help.
>Or is there somewhere more appropriate to go?
Online? hmmm... there might be chatrooms but linear algebra seems at a
level where a dedicated chatroom would be a little sparse. look for
undergrad math tutoring or help.
> I've always had
>trouble with vectors, and I think I fell apart sort of right at the
>beginning of linear algebra (although, I did manage to get a B- on the
>very first exam). I've got another exam next week. What can I do? I
>don't get all the terms, concepts, and jargon. Anyone know how to
>make learning linear algebra easier and more practical?
Practical? do you mean like showing how it is useful?
1) the contrite answer is that it is useful for fulfilling the
requirements of a degree in mathematics.
2) the liberal arts answer is that it is one of the pillars of
the beautiful and wonderful human achievement that is mathematics (along
with say differential/integral calculus, number theory, or logic). It
permeates all of mathematics.
3) the plain answer is that it is used all the time in engineering.
The answer you probably want though is that to make linear algebra easier
to understand (because it is all symbols, it's so abstract, you can lose
sight of what it really means) is to try to give it an interpretation that
is easier to think about. Usually geometrical/visual interpretations work
best with linear algebra. Think of arrows in the plane or 3-space as
vectors, a linear transformation (multiplying by a matrix) as modifying a
set of points, the determinant is what? eignevectors are what? similar
matrices behave how?) etc. ask your TA or prof about these...er after the
test.
>Anyone got any practice problems?
Online? possibly (google is your best bet here), but in real life, you can
go to a university library and find "Schaum's outlines" or "thousand's of
problems solved in" for linear algebra.
--
Mitch (remove the q to respond)
Marc Olschok
Actually, (simple) examples and theorems already help in the process
of learning the definitions. Of course a student needs to be willing
to look up the definitions in the text while going through these
examples.
The analogy with learning a foreign language has already been stressed:
although it is desirable that the vocabulary eventually "sinks in",
you would use new words in simple sentences early on in order to
foster memorization.
Marc
Jonathan Miller
news:c6e0nc$d3o$1...@news.math.niu.edu...
For linear algebra as she is usually taught, said minimum amount of talent
is pretty minimal.
Jon Miller
Felix Goldberg
Linear algebra is now "she"? That's news to me... :-)
Felix.
David Ames
> > Anyone got any practice problems?
>
> doesn't your book already have those?
Last I knew, there were problems at math.temple.edu
and by choosing "calculus on web" you got practice problems in other
lower-level math courses.
David Ames
Toni Lassila
>I think I may finally be getting a grip on what a vector space is:
>
>It's a group of vectors that can be multiplied by any scalar and/or
>added together in any way, and whatever possible combinations that can
>result is the "vector space" for that group of vectors.
Define "vector". You can't really, since you haven't properly defined
a vector space. Hint: axioms.
>understand it. For vectors in R^2, a plane is formed ("spanned"???)
>by the vector space. For vectors in R^3, a solid area is formed by
>the vector space.
Is span({0,0},{0,2}) a plane? Is span({0,0,0},{0,0,1},{0,0,2}) a solid
area?
--
"I'm not interested in mathematics that might have anything
to do with reality." -- Russell Easterly, in sci.math
Thomas Nordhaus
>ru...@vesuvius.math.niu.edu (Dave Rusin) wrote in message news:<c69sdb$oi7$1...@news.math.niu.edu>...
>> Our students "do well on the very first exam" because that's the part
>> of the course where we warm up with techniques for solving linear
>> systems of equations and such topics. But our LA course is also our
>> students' first course in which abstractions, axioms, and proofs play
>> a significant role. They often stumble because (among other problems)
>> they don't realize they need to _memorize_ definitions _precisely_.
>> So you can do a little self-assessment here to figure out whether
>> what you're missing is bits of topics or the core idea: can you,
>> right this minute, define what a vector space is?
>>
>> dave
>
>Dave, this may be my problem. I did decent on the first exam because,
>like you said, it was solving linear systems, echelon form, linear
>dependences--easier stuff like that. I've always been a little
>confused with the concepts, but I think I may finally be getting a
>grip on what a vector space is:
<Just some philosophizing:>
That's familiar: You feel comfortable with specific examples but can't
see the abstraction, the "structure" behind. In German there is a
saying meaning roughly: "You can't see the forest because of all the
trees"... As mathematics is the "science of structures", you will have
to get to this point. But it's a worthwhile goal to pursue because
it's really uplifting if you finally reach that point - As the old
greeks said: "Heureka!" I finally found!
Thomas
Thomas Nordhaus
>> then you don't know what a basis is. If you can't give the quantifiers
>> for the definition of independence, you won't be able to do a
>> proof using independence.
>>
>> --Dan Grubb
>
>Let me try that one...independence means a group of vectors (in
>homogenous form???)
Bad phrasing: "independence means a group of vectors". That would
translate: "Independence is a group of vectors...", doesn't make much
sence.
Usually you start definitions like: "A <set> is called <term to be
defined> if <this and that holds true>. Here:
(*) A <set of n vecors {v1,v2,...,vn}> is called <linear independent>
if...
>such that if they all equal the zero vector, then
No, if they all equal the zero vector they can't be linearly
independent.
>the only possible way for that is the each coefficient of every vector
>has to equal 0 too.
OK, that sounds better. But what is "that"? And what is "each
coefficient"? You haven't used or mentioned any coefficient yet. Here
is a way of phrasing it:
(* continued): "... given any coefficient c1,c2,...,cn ..." (you have
to give those things a name!) "... c1*v1+c2*v2+...+cn*vn = 0 ..."
(that's the "that") "... implies c1=c2=...=cn=0"
So, now you have a formal definition!
Thomas
George Cox
>
> ru...@vesuvius.math.niu.edu (Dave Rusin) wrote in message news:<c69sdb$oi7$1...@news.math.niu.edu>...
> > right this minute, define what a vector space is?
> >
> grip on what a vector space is:
>
> added together in any way, and whatever possible combinations that can
> understand it. [...]
Then you're wrong. This waffle is next to useless. What is the precise
definition? If the precise definition has words like "abelian group" in
it, then what is the precise meaning of them? And so on.
Read David C Ullrich's reply to your op, it's good advice. (Which will
also apply to other pure mathematics courses.) Having memorized the
definitions, begin to understand them by constructing examples for
yourself. The true understanding will come when you read proofs of
theorems and appreciate that the definitions were phrased in just the
way they are so as to make the theorems true.
Also, what book do you read? If it doesn't have good hard exercises in
it, then read another.
--
G.C.
Note ANTI, SPAM and invalid to be removed if you're e-mailing me.
David C. Ullrich
>gr...@lola.math.niu.edu (Daniel Grubb) wrote in message news:<c6bars$ejg$1...@news.math.niu.edu>...
>> [...]
>>
>> The student should be able to give a rendition that is at least
>> equivalent to the one given in the book and that uses precise language.
>> If you can't say that a basis is an independent spanning set
>> then you don't know what a basis is. If you can't give the quantifiers
>> for the definition of independence, you won't be able to do a
>> proof using independence.
>>
>> --Dan Grubb
>
>Let me try that one...independence means a group of vectors (in
>homogenous form???) such that if they all equal the zero vector, then
>the only possible way for that is the each coefficient of every vector
>has to equal 0 too.
Nope. You're illustrating precisely why you're failing the class.
That's _not_ the definition.
Actually from what you say here it seems pretty likely that
you do know what linear independence _is_, but the
way you're stating the definition is totally wrong. Knowing
something doesn't help much if you can't explain it
coherently. Next time you take the class try actually
_learning_ the definitions of all the important terms,
_precisely_. It _will_ help, whether you believe it or not.
************************
David C. Ullrich
David C. Ullrich
>[...]
>
>I've come a long way in my understanding of linear algebra using the
>guides I have, but, like I've said, I'm still really unsure what my
>problem is.
It's perfectly clear to me what the problem is. It's exactly what
I guessed, just hearing that you were doing poorly in linear
algebra - seeing your attempt at defining the two terms
"independent" and "vector space" just now I'm certain my
guess was correct.
Your problem is you don't know the defintions of the words.
You think you know what the words mean, you don't appreciate
that you need to know the _exact_ definitions, _precisely_.
There's nothing anyone can do to help you learn the definitions,
you simply have to _do_ it. _After_ you know the definitions
people can help you understand what they mean and how
they're used. But _first_ you have to _learn_ them.
> Something tells me I may just need a little more brushing
>up on a little more specific useage of the concepts, and I may be OK.
>Finding a way to solve proofs might be helpful too; this is something
>I generally suck at and not having all the concepts clearly seems to
>only make the proofs harder.
Not having the concepts clear makes the proofs _impossible_.
The next time you want to explain what linear independence means
don't say what you said in that post just now, instead say
"vectors v_1, ... v_n are independent if the only solution to
c_1 v_1 + ... + c_n v_n = 0 is c_1 = c_2 = ... = 0." You think
that's the same as what you said. It may well be the same
as what you meant, but it's _not_ the same as what you
_said_ - if you could state the definition _correctly_ you'd
have a _chance_ with the proofs.
************************
David C. Ullrich
David C. Ullrich
>David C. Ullrich <ull...@math.okstate.edu> wrote in message news:<ipth80lfrcm14ija6...@4ax.com>...
>> On 22 Apr 2004 15:44:12 -0700, de...@aol.com (DE781) wrote:
>>
>> >I'm currently a math major and am taking linear algebra, but I'm in
>> >serious danger of failing. I just don't get it! Is this newsgroup a
>> >place to come to ask questions and get information about learning
>> >math? Or is there somewhere more appropriate to go? I've always had
>> >trouble with vectors, and I think I fell apart sort of right at the
>> >beginning of linear algebra (although, I did manage to get a B- on the
>> >very first exam). I've got another exam next week. What can I do? I
>> >don't get all the terms, concepts, and jargon. Anyone know how to
>> >make learning linear algebra easier and more practical? Anyone got
>> >any practice problems?
>
>Thanks for the post. I've read it all the way through and agree and
>understand what you're saying. About the definitions, I like your
>definition for "basis": "independent spanning set" because it's short
>and simple. Therefore, it's *easy* to memorize. My text book would
>tend to defend the basis in a really, really abstract way like: "Let S
>[contained in symbol] V be a subset of V. If *x* [element symbol] S =
>a1x1 + a2x2 + ... + a^nx^n = *0*, then *x* is a basis if a^i (1 < i <
>n) = 0 for all a^i and Span(V) = S.", or something really wordy and
>convoluted like that. I mean, I eventually understand what the
>definition is saying. But, "independent spanning set" is just so much
>easier, IMO.
Glad you liked the definition. The only reason it seemed simpler
than the definition in the book is that I didn't define the words
"independent" and "spanning".
Here's the deal with definitions. But it only works if you know
them _exactly_ right, including the wordy convoluted ones.
If you take "independent spanning set" and insert the
defintions of "independent" and "spanning" you'll get the
definition of "basis" in the book.
************************
David C. Ullrich
rus...@mdli.com
>ru...@vesuvius.math.niu.edu (Dave Rusin) wrote in message news:<c69sdb$oi7$1...@news.math.niu.edu>...
>> In article <c98b1ba0.04042...@posting.google.com>,
>> DE781 <de...@aol.com> wrote:
>> >I'm currently a math major and am taking linear algebra, but I'm in
>> >serious danger of failing. I just don't get it! Is this newsgroup a
>> >place to come to ask questions and get information about learning
>> >math? Or is there somewhere more appropriate to go? I've always had
>> >trouble with vectors, and I think I fell apart sort of right at the
>> >beginning of linear algebra (although, I did manage to get a B- on the
>> >very first exam). I've got another exam next week. What can I do? I
>> >don't get all the terms, concepts, and jargon. Anyone know how to
>> >make learning linear algebra easier and more practical? Anyone got
>> >any practice problems?
>>
>> I would bet that the single practice problem you need to work on is,
>> "What is a vector space?"
>>
>> Our students "do well on the very first exam" because that's the part
>> of the course where we warm up with techniques for solving linear
>> systems of equations and such topics. But our LA course is also our
>> students' first course in which abstractions, axioms, and proofs play
>> a significant role. They often stumble because (among other problems)
>> they don't realize they need to _memorize_ definitions _precisely_.
>> So you can do a little self-assessment here to figure out whether
>> what you're missing is bits of topics or the core idea: can you,
>> right this minute, define what a vector space is?
>>
>> dave
>
>Dave, this may be my problem. I did decent on the first exam because,
>like you said, it was solving linear systems, echelon form, linear
>dependences--easier stuff like that. I've always been a little
>confused with the concepts, but I think I may finally be getting a
>grip on what a vector space is:
>
>It's a group of vectors that can be multiplied by any scalar and/or
>added together in any way,
Not just any way -- there are precise conditions that the sum must
satisfy. Look for the axiomatic definition in your book(s). It's not
at all difficult to memorize. Do it.
and whatever possible combinations that can
>result is the "vector space" for that group of vectors. This is how I
>understand it. For vectors in R^2, a plane is formed ("spanned"???)
>by the vector space. For vectors in R^3, a solid area is formed by
>the vector space. It gets difficult for me to move into dimension 4.
Geometric examples are often helpful, but use them to *liberate*
yourself, not tie yourself down. If you are faced with a vector in
R^4 on your exam, don't bother trying to visualize it. You can do
everything you need to do algebraically.
Personally, I keep a few geometric examples from R^2 in my head, and
don't bother with anything more complicated unless the problem happens
to be specifically about geometry, which won't be the case on your
exam this week. E.g. the matrix for a 90-degree rotation in R^2, in
the usual basis, is useful to remember -- or even better, you might
learn how to derive it quickly on the fly -- because if (say) you find
yourself confused about how basis vectors transform, you have a ready
example to check out and remind yourself. Other 2x2 matrices for
simple rotations and reflections (and perhaps more) are good to know,
but not for their own sakes, but rather because they help you think
clearly about the abstractions (and only if they truly do).
>While I understand that the same concepts hold, there's no more
>physical picture I can use to visualize what's happening. Is my
>understanding of "vector space" sufficient enough? Am I missing
>anything?
The algebra is the important thing, not the picture. And your current
understanding of the algebra is insufficient to keep you from getting
confused on the upcoming test. You need to learn the definitions
precisely.
>
>I know I am still struggling with the concepts of "span" and "basis".
>The weird thing is that I'm alright with the more advanced stuff;
>matrices, determinants, eigenvalues, eigenvectors. I'm a little hazy
>with diagonalization because it's the newest thing we've done. I know
>it's got something to do with the eigenvalues of a special type of
>matrix.
>
>I guess I could also really use some help with understanding how a
>mapping gets converted into a matrix, and then how to solve it.
I like the term "linear transformation" and I think you should use it
too; a "mapping" usually means something more general that may not
have the necessary restrictions. Definition time again! Do you know
what restrictions I'm talking about? f(a+b) = f(a)+f(b) and
f(ca) = cf(a) of course. That, by definition, is what makes the
mapping *linear*.
Anyhow, you ask about how a linear transformation is converted into a
matrix. The l.t. is the abstract thing; the matrix is one particular
representation of it (in terms of two bases). Change one or both of
the bases, and you get a different matrix for the same linear
transformation. It's somewhat analogous to the way you can write the
same abstract number (say, the cube of two) as 8 in decimal, or 10 in
octal, or 1000 in binary. But better than that, I think, because all
of the important features of the linear transformation have
counterparts in matrix theory; i.e. whatever matrix you end up using,
it will share some important features with the transformation, e.g. it
will have the same eigenvalues, same rank, etc.
(So, one can say that the set of nxm matrices is isomorphic to the set
of linear transformations from F^n to F^m. Indeed this is an
isomorphism in the strict sense I give below, because both these sets
are vector spaces in their own right! But don't worry about that if
you find it confusing.)
In answer to your question, you aren't *really* converting the
transformation into a matrix -- they are two different things -- but
for most practical purposes you can conveniently ignore that fine
distinction
OTOH if what you're asking is how (by what method) to do this
conversion, that actually is quite easy. Take basis vector #1, apply
the transformation to it componentwise to get the components of the
transformed vector, and write those components in a column. Do the
same with basis vector #2, writing it as a column to the right of the
one you already wrote. And so on, until you're done. There's your
matrix. Try working it out for the 90-degree rotation I mentioned
above, and then try your matrix out on some 2D column vectors to see
if they really do turn 90 degrees when you multiply by the matrix.
(I am assuming you multiply with matrix on the left and column vector
on the right, as is done in Schaum's outline; hopefully that is how
your prof does it too, otherwise swap positions and take the
transposes!)
>I
>understand matrix multiplication and can do it well. But the concepts
>of image, kernel, and isomorphism and how they relate to the
>mappings/matrices seem to be lost on me. The odd thing is that I
>fully understand the definitions of "kernel" and "image" as they were
>applied in algebraic structures, but I don't get how they apply to
>linear really. "Isomorphism" is a concept I never understood in
>algebraic structures or linear algebra.
If you get it for algebraic structures in general, you are ahead of
the game. In the context of linear algebra, an isomorphism is simply
a linear transformation (remember the definition?) that has an
inverse. Kernel and image have their usual meaning.
(Isomorphisms come up a lot in math; see above. In general they tell
you when it's OK to substitute one structure by another, without
losing any essential information.)
One nice thing about linear algebra is that once you have a matrix for
a linear transformation, you can answer questions e.g. about the
l.t.'s kernel and image very easily, simply by manipulating the
matrix, without having to wrestle with the abstraction. If the kernel
is {0}, i.e. the set of rows (or the set of columns) of the matrix is
linearly independent, and the matrix is square, then the matrix is
invertible and you have an isomorphism. Otherwise, one or more rows
or columns is a linear combination of the others; and that means there
are nonzero vectors which, when multiplied by your matrix, give zero.
In other words, those vectors are members of the kernel of your linear
transformation. See how it all fits together?
Btw you asked elsewhere what an eigenvalue is. The definition is
simplicity itself, learn it! If your question is really, what are
they good for, one answer is that they come up a lot in differential
equations. A part of math that linear algebra has a lot to say about.
>
>Thanks to the people who posted book suggestions, but I'm hesitant
>about buying any other books. I already bought the text, the Cliff's
>Notes guide to linear algebra, a 2003-version of the Schaum's outline,
>and I even have an old 1968 version of Schaum's that my grandmother
>used when she majored in math. Cliff's has been helpful, but too
>basic. Schaum's seems almost too advanced; it's great that they solve
>all the problems, but sometimes the explanations are lacking. I find
>that I do much better at math problems if I can first figure out how
>to solve a certain type of problem and then go back and try to
>understand the concepts behind it, rather than the other way around.
>Schaum's examples don't allow for this, because they assume you've
>already read (and understood) the concepts behind how to solve certain
>problems.
Yes, I think you have enough books. Personally, I like the Schaums
Outline (the standard one by Lipschutz) a lot, by the way. As one who
easily gets confused myself, I can say it was the book that made the
quickest sense to me, i.e. had the smallest eyes-glaze-over effect.
OTOH, I did have the benefit of having read and worked on some other,
more abstract books beforehand. Btw I found nothing objectionable
about Lipschutz from an abstract point of view; if he made it any
simpler he would mislead, but he treads that fine line very nicely
IMHO.
>
>Maybe, if it isn't too much to ask, would anyone here be willing to
>post some problems relating to mappings/kernel/image/isomorphims
>and/or eigenvalues/eigenvectors, and I can attempt to solve them with
>your help?
Learn the definitions first, vet them with some examples and
counterexamples as some others have recommended, and then focus on
learning some important techniques cold, and with luck you can ace the
test (and be a math major too, and even be happy about it someday).
Usenet will probably be too slow for your needs this week; who wants
to type in those matrices in ASCII anyway.
Thomas Nordhaus
>Thomas:
>
>>You haven't used or mentioned any coefficient yet. Here
>>is a way of phrasing it:
>>
>>(* continued): "... given any coefficient c1,c2,...,cn ..." (you have
>>to give those things a name!) "... c1*v1+c2*v2+...+cn*vn = 0 ..."
>>(that's the "that") "... implies c1=c2=...=cn=0"
>>
>>So, now you have a formal definition!
>
>But, that's exactly what I said, even if my wording made it a little confusing.
No, you said something different and - frankly - incoherent. It is not
verifyable and falsifiable. And one phrasing was even wrong: "...such
that if they all equal the zero vector..."
> I knew that this was the definition, so you can't say I don't know the
>definition of "independence".
But I can't say that you know, either.
>I may not know how to phrase it exactly, but I
>know what an "independent set" is. Why should such a precise definition be
>needed for computation? (I can understand for proofs, but I gotta work out
>computational mistakes first.)
You can prove that by solving practical problems, that's a different
story.
Thomas
Marc Olschok
> David:
>
>>Actually from what you say here it seems pretty likely that
>>you do know what linear independence _is_, but the
>>way you're stating the definition is totally wrong.
>
> gets me. It's like this, probably, for a bunch of the main terms. Still, you
> say the number one problem is that students don't understand all the
> definitions. Is my understanding of "independence" then not good enough? Or
> is it?
A definition may be phrased differently in different texts. But these
variatons should be eqivalent. Starting with one of these variants with
the *exact* phrasing you will usually build an internal mental representation.
You should be able to reproduce the original definition from this
internal representation. If this does not work (i.e. "the phrasing gets you"),
this indicates that something went wrong along the way.
The problem is that you will confuse yourself each time you need to know
the original concept, e.g. in a proof.
I suggest the following steps when confronted with the definition of
a new concept X:
1. While reading the definition of X for the first time
(a) ignore previous encounters with anything that just "looks like" X.
(b) if the definition of X uses concept Y then lookup the definition
of Y unless you are able to reproduce it yourself.
for example, when you encounter the definition of "vector space",
ignore that you might met the term "vector" before.
2. after having read the definition of X look for (non)examples of X.
at this stage, _do_not_hesitate_to_look_at_the_definition_of_X_again.
Do not count on being able to memorize the definition of X just from
reading it once or twice. It is far worse to memorize a distorted version
of the original definition.
Working on examples this way will eventually help you to memorize the
definition.
3. look at proofs of statements that involve concept X. Try it yourself.
This will help you to memorize X.
4. After a while check if you can reproduce the original definition of X.
The main point in steps 2 and 3 is to avoid the fatal sequence
"read X once , remember X' instead, use X' until it sinks in,
never really understand X"
For this it is crucial that you check constantly whether you really are
really using the correct definition.
>
>> Next time you take the class try actually
>>_learning_ the definitions of all the important terms,
>>_precisely_.
>
> OK, there's my answer, I guess. I've always had courses where professors
> emphasized that we don't have to memorize definitions and terms word for word,
> whether it's a foreign language course, a social studies course, an economics
> course, whatever.
They probably just meant to stress, that you should not _stop_ at
memorizing word for word. Also they may have counted on previous
knowledge about the concepts taught. But I guess, you learned at
least foreign vocabulary word for word.
> It seems like definitions are usually just things that you
> have to get the basic jist of. It's sort of unfair that math is different and
> no one warns us about it.
Really?
Marc
David C. Ullrich
>David:
>
>>Actually from what you say here it seems pretty likely that
>>you do know what linear independence _is_, but the
>>way you're stating the definition is totally wrong.
>
>Right. That's my point. I *do* know the definition, but the phrasing always
>gets me.
Aargh. If the phrasing gets you then you _do_ _not_ know the
definition!
> It's like this, probably, for a bunch of the main terms. Still, you
>say the number one problem is that students don't understand all the
>definitions. Is my understanding of "independence" then not good enough?
If you can't state the definition coherently, which indeed you can't,
then no, your understanding is not good enough to be able to
write proofs.
Really. A feel to nice for how. To get it, so if you have to ride then
to it, otherwise. So wit unless rain time less.
(What, you didn't follow that last paragraph? I was trying to
say "it's a nice day here in Oklahoma". I know what I meant,
it's the phrasing that always gets me...)
> Or
>is it?
>
>> Next time you take the class try actually
>>_learning_ the definitions of all the important terms,
>>_precisely_.
>
>OK, there's my answer, I guess. I've always had courses where professors
>emphasized that we don't have to memorize definitions and terms word for word,
>whether it's a foreign language course, a social studies course, an economics
>course, whatever. It seems like definitions are usually just things that you
>have to get the basic jist of. It's sort of unfair that math is different and
>no one warns us about it.
Math _is_ _very_ different in this regard. I certainly warn my
students of this, rarely helps. Your professors are just assuming
that you _realize_ that you need to know what the words mean...
> But, like I said, I won't have the opportunity to
>try linear algebra your way because if I fail, it's my last math course ever.
>I think there may still be hope for me....
************************
David C. Ullrich
David C. Ullrich
(Arturo Magidin) wrote:
>In article <20040427182710...@mb-m18.aol.com>,
>DE781 <de...@aol.com> wrote:
>>David:
>>
>>>Actually from what you say here it seems pretty likely that
>>>you do know what linear independence _is_, but the
>>>way you're stating the definition is totally wrong.
>>
>>Right. That's my point. I *do* know the definition, but the phrasing always
>>gets me.
>
>That tells me that you do not really "know" the definition. You
>->think<- you know it, you ->think<- you understand it, but you
>actually do not. If you did, the phrasing would not be a problem.
Or, as seems quite possible from the definition he gave, he
does understand what independence means, but he doesn't
understand how to state things coherently. (Which of course
is equally fatal - one of many reasons for memorizing these
definitions vergatim is to give us a stock of examples to
use in learning _how_ to say _exactly_ what we mean.)
>> It's like this, probably, for a bunch of the main terms. Still, you
>>say the number one problem is that students don't understand all the
>>definitions. Is my understanding of "independence" then not good enough? Or
>>is it?
>
>If your understanding is not sufficient to lead you to a coherent and
>correct statement of the term, then it is not good enough. Unless you
>can state coherently and correctly what the definition is, then it
>will cause you problems when you try to use it.
************************
David C. Ullrich
David C. Ullrich
>Toni:
>
>>Define "vector". You can't really, since you haven't properly defined
>>a vector space. Hint: axioms.
>
>I don't understand. A vector is any collection: (x1, ...., x^n) of anything.
>In math, the vectors are numbers.
Absolutely not. (1,2,3) is an example of a vector in a certain vector
space. But what "vector" actually means is "element of a vector
space", and what "vector space" actually means is [insert
longish definition here].
An example I always give to emphasize that a vector space
is exactly what the definition says it is, not what we think
it is:
Let V = {the movie "Kill Bill"}. Let's say M is that movie, to
save typing; now V = {M}.
For x, y in V define x + y = M. For x in V and a real number
r, define fx = M.
Now it's easy to verify (_if_ you _know_ the definition of
"vector space" it's easy, anyway) that V, together with
the addition and scalar multiplication defined above,
is a vector space. So now the movie "Kill Bill" has
become a vector.
(Note it's the movie itself that's a vector, not (the movie)
or the sequence of frames of the movie or whatever.)
>>Is span({0,0},{0,2}) a plane?
>
>No, because a1 (0,0) + a2 (0,2) = 2a2. Multiplied by any scalar, this gives
>you a line through the origin and 2a2. So, it's just a line. I guess I should
>have said any vector space in R^2 is at MOST a plane?? Or, maybe, any basis
>for R^2 is a plane. Right? Since the (0,0) part of the span above is not
>needed. So, the above span is not independent and is therefore not a basis.
>That's why you just get a line.
>
>> Is span({0,0,0},{0,0,1},{0,0,2}) a solid
>>area?
>
>Again, this would be a line (????): 3a3. Only either (0,0,1) or (0,0,2) is
>needed to produce a basis.
************************
David C. Ullrich
Marc Olschok
> George:
>
>>Then you're wrong. This waffle is next to useless. What is the precise
>>definition? If the precise definition has words like "abelian group" in
>>it, then what is the precise meaning of them?
>
> The only time we spoke of abelian groups was in algebraic structures. If I
> remember correctly, an abelian group is a group that has an invertible table.
What is an "invertible table" ? By the way, a group is abelian if the
group operation is commutative.
You can group the axioms in the definition for a vectorspace as follows
(a) those axioms only involving addition
(b) those axioms only involving multiplication with scalars
(c) the mixed distributivity laws
those axioms in (a) are those for an abelian group.
In most linear algebra courses, the definition of an (abelian) group
should at least be mentioned because
- the axioms from (a) can be remembered easier.
- the definition of a field also includes them for the field addition.
- a lot of concepts from the theory of vectorspaces have counterparts
in the theory of abelian groups; this gives a chance for learning
through (modified) repetition.
Marc
Grey Knight
> har...@tcs.inf.tu-dresden.de (Mitch Harris) wrote in message news:<c6e1ep$aov3s$1...@uni-berlin.de>...
[NB: I write <<M>> for matrices, <x> for vectors, <<A'>> for the
inverse of A, <<A+>> for its transpose and det(A) for its determinant.
<<I>> represents the identity.]
> >the determinant is what?
Some interesting facts about determinants:
det(<<A>> <<B>>) = det(<<A>>) det(<<B>>)
If one row or column of <<A>> consists of all zeroes, det(<<A>>)=0
If any two rows or columns of <<A>> are identical, det(<<A>>)=0
det(a <<A>) = a^n det(<<A>>) for n-by-n matrices
det(<<A'>>) = 1/det(<<A>>) provided det(<<A>>) not= 0 in which case
<<A'>> doesn't exist
det(<<A+>>) = det(<<A>>)
det(<<X>> <<A>> <<X'>>) = det(<<A>>) provided det(<<X>>) not= 0
>
> >eignevectors are what?
> If T(*x*) = (constant)(*x*), then (*x*) is the eigenvector and
> (constant [lambda]) is the eigenvalue, right?
A matrix <<A>> represents a linear transformation; call its
eigenvalues l_i and the corresponding eigenvectors <x_i>. Then any
point which can be written a*<x_i> will transform to the point
(l_i)*a*<x_i>; that is, points starting on the line parallel to <x_i>
through the origin will stay on that line. The eigenvalue tells you
how they move along it. Interestingly, If you apply the transformation
<<A>> to any vector <x> a large number of times, the result tends to
become parallel to the eigenvector whose corresponding eigenvalue has
the largest absolute value (ie: the <x_i> such that |l_i| is greater
than for all other l_j).
> I still need to understand what has to be done to solve for the eigenvalues
> and eigenvectors though.
If you calculate det(<<A>>-l*<<I>>) for an n-by-n matrix <<A>>, you
will get a degree-n polynomial in l; generally you'll be expected to
deal with 2-by-2 matrices, so this polynomial is just a quadratic. The
solutions of this polynomial are your eigenvalues. The eigenvectors
can then be found by solving the problem <<A>> <x_i> = l_i <x_i>; that
is (<<A>> - l_i<<I>>)<x_i> = <0>
(This is where the expression det(<<A>>-l*<<I>>) for the eigenvalues
comes from, BTW)
Interestingly, if you construct a matrix <<L>> = diag(l_i) from the
eigenvalues (that is, the diagonal elements of <<L>> are the l_i; the
other elements are zero) and another matrix <<X>> by composing the
eigenvectors as columns (that is, each column of <<X>> is one of the
eigenvectors), making sure to get the eigenvalues in the same order as
the eigenvectors, then
<<A>> = <<X>> <<L>> <<X'>>
which is a similarity transformation! <<L>> is much easier to work
with than <<A>>, so this pays off greatly. In particular you can
compute powers easily:
<<A>> ^ m = <<A>> <<A>> ... <<A>>
= (<<X>> <<L>> <<X'>>) (<<X>> <<L>> <<X'>>) ... (<<X>> <<L>>
<<X'>>)
= <<X>> <<L>> (<<X'>> <<X>>) <<L>> (<<X'>> ... <<X>>) <<L>>
<<X'>>
= <<X>> <<L>> <<I>> <<L>> <<I>> ... <<I>> <<L>> <<X'>>
= <<X>> (<<L>> ^ m) <<X'>>
which is far easier to calculate; since <<L>> is diagonal, <<L>>^m
just means you raise each of the diagonal elements to the m-th power
separately!
> similar
> > matrices behave how?)
> I don't know what a similar matrix is. Is that where one of them
> equals the other times a scalar? If so, how does one determine if two
> matrices *are* similar?
Two matrices are similar iff
<<A>> = <<X>> <<B>> <<X'>>
where <<X>> must obviously be nonsingular (ie: det(<<X>>) not= 0),
otherwise <<X'>> doesn't even exist!
If <<B>> represents some linear transformation with basis vectors
<e_i>, then <<A>> represents the same transformation with the basis
vectors <d_i> = <<X>> <e_i>
Tim Mellor
> in article <c6bekt$toj$1...@news.math.niu.edu>,
> daniel grubb <gr...@math.niu.edu> wrote:
>
> |>|>Memorize? I can't remember ever memorizing anything. Better just to
> |>|>practice until you understand. Discuss, ask questions, apply. That way
> |>|>you memorize, of course, but that's just a side-effect.
> |>|
> |>|No, for proof classes, which linear algebra is in many places,
> |>|it is crucial to *memorize* the definitions.
>
> |>nonsense.
> |
> |I'm afraid not.
>
> in that case you ought to have at least tried to give some reason to
> support your opinion.
Here's one. The first question in the exam is likely to be something
like the following:
1. Give the definition of a vector space.
Or perhaps
2. Let X be ....
Prove that X is a vector space.
(How is this done? By checking all the conditions that appear in the
definition of "vector space", so again one needs to be able to write
this down.)
Tim Mellor
> David:
> OK, there's my answer, I guess. I've always had courses where professors
> emphasized that we don't have to memorize definitions and terms word for word,
> whether it's a foreign language course, a social studies course, an economics
> course, whatever. It seems like definitions are usually just things that you
> have to get the basic jist of. It's sort of unfair that math is different and
> no one warns us about it. But, like I said, I won't have the opportunity to
> try linear algebra your way because if I fail, it's my last math course ever.
> I think there may still be hope for me....
I think this is an important point. Maths is different from other
academic diciplines in a number of ways, but this is often not
stressed enough, and further it is difficult to get people to
understand the differences.
In particular mathematics requires precision, you must use the precise
definition and statements of theorems. If you do not, the results may
be startlingly different.
As an aside, I have found that many undergraduates do think that maths
is different to other subjects in the following way. They think that
because one is making use of symbols, numbers et cetera, one need not
write sentences. Thus they give a list of equations as a proof with no
indication about how they beleive these equations are linked. It can
ofter take many months, repeated explinations and many zeros for
coursework before changing this concept. Grrr...
Daniel Grubb
>>Actually from what you say here it seems pretty likely that
>>you do know what linear independence _is_, but the
>>way you're stating the definition is totally wrong.
>Right. That's my point. I *do* know the definition, but the phrasing always
>gets me. It's like this, probably, for a bunch of the main terms. Still, you
>say the number one problem is that students don't understand all the
>definitions. Is my understanding of "independence" then not good enough? Or
>is it?
As others have said, if the phrasing is a problem, then you don't
understand the definition. That can be because you don't have the concept,
or because you can't say things coherently. The problem is that, to
do math, you *have* to say things coherently. Precise language is
crucial. Fuzzy concepts are simply not good enough. Let's look at your
definition.
>Let me try that one...independence means a group of vectors (in
>homogenous form???) such that if they all equal the zero vector, then
>the only possible way for that is the each coefficient of every vector
>has to equal 0 too.
I'm going to pick this apart so you can see how language gets used.
>independence means a group of vectors
No. 'We say that a set of vectors is independent when....'
As you said it, you imply that independence *is* a set of vectors, but
it isn't. A set of vectors can have the property of independence. We will
be defining what it takes for a collection of vectors to have that
property.
>(in homogenous form???)
This certainly suggests some lack of understanding. The set of vectors
is in homogeneous form? A *set* cannot be in homogeneous form. An
algebraic expression can be. Be careful exactly what it is that has a form.
>such that if they all equal the zero vector
All the vectors are equal to 0? Are you sure you want each and every one
of the vectors to be 0? In other words, there is only one vector in the
set? As you have said it, you mean that the only independent set of
vectors is the set with only the zero vector. Is that what you want?
I suspect not.
>then
>the only possible way for that is the each coefficient of every vector
>has to equal 0 too.
What coefficient? You didn't mention or introduce any coefficients! You
only have some vectors which are all 0!
Let's look at a real definition:
A set of vectors {v_1 , ...v_n} is independent if the only way for
a linear combination c_1 v_1 +...c_n v_n to be the 0 vector is when
c_1 = ...=c_n=0.
Some things to notice:
*If* c_1 = ...=c_n =0, then for *any* collection of vectors {v_1, ..v_n},
independent or not, c_1 v_1 +...c_n v_n=0. Do you see why? So it is
possible for a linear combination to be 0, yet the vectors in
that linear combination not to be independent.
The definition is precise and checkable: If you can find c_1,...,c_n
that are not all 0 such that c_1 v_1 +...c_n v_n =0, then the set
{v_1 ,...v_n} is *not* independent. Do you see why? If you can show
that no such collection of c_i exists, then the set *is* independent.
Do you see why?
More importantly, do you see the difference between your definition and
the one I gave?
--Dan Grubb
James Dolan
tim mellor <ti...@amsta.leeds.ac.uk> wrote:
|jdo...@math-cl-n01.math.ucr.edu (James Dolan) wrote in message
|news:<c6bfjv$9vn$1...@glue.ucr.edu>...
|> in article <c6bekt$toj$1...@news.math.niu.edu>,
|> daniel grubb <gr...@math.niu.edu> wrote:
|>
|> |>|>Memorize? I can't remember ever memorizing anything. Better just to
|> |>|>practice until you understand. Discuss, ask questions, apply. That way
|> |>|>you memorize, of course, but that's just a side-effect.
|> |>|
|> |>|No, for proof classes, which linear algebra is in many places,
|> |>|it is crucial to *memorize* the definitions.
|>
|> |>nonsense.
|> |
|> |I'm afraid not.
|>
|> in that case you ought to have at least tried to give some reason to
|> support your opinion.
|
|Here's one.
no. your "reason" is essentially just that some total moron might
threaten to punish you unless you memorize the definitions. you could
as easily use such a "reason" to justify any arbitrarily stupid
behavior.
|The first question in the exam is likely to be something
|like the following:
|
|1. Give the definition of a vector space.
|
|Or perhaps
|
|2. Let X be ....
|Prove that X is a vector space.
|
|(How is this done? By checking all the conditions that appear in the
|definition of "vector space", so again one needs to be able to write
|this down.)
--
[e-mail address jdo...@math.ucr.edu]
Daniel Grubb
>|Here's one.
>no. your "reason" is essentially just that some total moron might
>threaten to punish you unless you memorize the definitions. you could
>as easily use such a "reason" to justify any arbitrarily stupid
>behavior.
Not arbitrary or stupid.
>|The first question in the exam is likely to be something
>|like the following:
>|
>|1. Give the definition of a vector space.
>|
>|Or perhaps
>|
>|2. Let X be ....
>|Prove that X is a vector space.
It is certainly reasonable for a linear algebra exam to
have a student show that a certain structure is a vector space!
--Dan Grubb
James Dolan
daniel grubb <gr...@math.niu.edu> wrote:
how do you manage to memorize mathematical definitions when you don't
even seem to be able to remember the issue under discussion in a
newsgroup thread? the issue was whether there is any reason to
memorize definitions by rote instead of just using a piece of paper or
computer to memorize them for you, so your comment is entirely
irrelevant.
--
[e-mail address jdo...@math.ucr.edu]
Dave Rusin
in Linear Algebra is his or her ability to define things accurately.
Others have responded that memorization per se is useless, that
strict memorization is unnecessary, or that learning the definitions
is not getting at the heart of the subject; and I agree that, taken by
themselves, these statements are correct. Still, I hold to the stance
in my first paragraph because, for the average student at my institution,
memorizing the definitions is the way to begin this course.
Now, in article <c98b1ba0.04042...@posting.google.com>,
DE781 <de...@aol.com> wrote:
>>>> So you can do a little self-assessment here to figure out whether
>>>> what you're missing is bits of topics or the core idea: can you,
>>>> right this minute, define what a vector space is?
>>
>>> Hopefully not just as a one-to-one rendition of phrases from a
>>> textbook...
>>
>> The student should be able to give a rendition that is at least
>> equivalent to the one given in the book and that uses precise language.
>
> Let me try that one...independence means a group of vectors (in
> homogenous form???) such that if they all equal the zero vector, then
> the only possible way for that is the each coefficient of every vector
> has to equal 0 too.
OK, now, jdolan and others who pooh-poohed the idea of memorizing
definitions: what say you to this student? Seems to me he has
made my point for me ...
dave
PS -- This student's definition is no worse than the average that I
would get here from a student preparing for our final exam. Not
surprisingly, there are not many A's and B's when I give semester grades.
Dave Rusin
George Cox <george_...@SPAMbtinternet.com.invalid> wrote:
[Vector space.]
>What is the precise
>definition? If the precise definition has words like "abelian group" in
>it, then what is the precise meaning of them? And so on.
Yeah, sure. A vector space is a module over a field. That's absolutely
true and probably given that way in Bourbaki somewhere, but even in
selective institutions I can't believe students learn about modules
before vector spaces. Is there really a source anywhere that pretends to
_introduce_ vector spaces in terms of abelian groups?
Hmm, I suppose some here would argue that it messes up students'
understanding of modules if they are first tainted by facts learned
for vector spaces, which then have to be unlearned in the general case.
dave
Dave Rusin
David C. Ullrich <ull...@math.okstate.edu> wrote:
>An example I always give to emphasize that a vector space
>is exactly what the definition says it is, not what we think
>it is:
>
>Let V = {the movie "Kill Bill"}. Let's say M is that movie, to
>save typing; now V = {M}.
Right. Every time I teach Linear Algebra, I point out that, among my
many accomplishments and titles, I am the unique member of a Vector Space.
(You too can be a vector space. Act Now! For only $5.95, ...)
dave
PS -- The clever students have pointed out that this means I am
the Zero Vector. I don't put that on my resume.
Dave Rusin
>I already bought the text, the Cliff's
>Notes guide to linear algebra, a 2003-version of the Schaum's outline,
>and I even have an old 1968 version of Schaum's that my grandmother
>used when she majored in math.
This is not your grandmother's Linear Algebra course!
(I've always wanted to say something like that.)
>Maybe, if it isn't too much to ask, would anyone here be willing to
>post some problems relating to mappings/kernel/image/isomorphims
>and/or eigenvalues/eigenvectors, and I can attempt to solve them with
>your help?
Sample question
(a) Prove that the set M of all n by n matrices is a vector space (using
familiar matrix addition and scalar multiplication.) What is its
dimension?
(b) Prove that the map f(x) = x^t is a linear transformation from M to M
What is its kernel?
(c) Compute the eigenvalues of f and find the eigenspaces.
I am of course deliberately choosing a question which emphasizes the
proper use of terminology and abstraction, but this is a perfectly
reasonable exam question. (IMHO -- but I have a reputation for thinking
"interesting" questions are reasonable so maybe you shouldn't trust me.)
dave
Daniel Grubb
>|It is certainly reasonable for a linear algebra exam to
>|have a student show that a certain structure is a vector space!
>how do you manage to memorize mathematical definitions when you don't
>even seem to be able to remember the issue under discussion in a
>newsgroup thread? the issue was whether there is any reason to
>memorize definitions by rote instead of just using a piece of paper or
>computer to memorize them for you, so your comment is entirely
>irrelevant.
No, I did remember the context. I *do* think it is reasonable
for a student to show a certain structure is a vector space
without having a machine or piece of paper with the definition
on it. If they cannot show that, say, the collection of anti-symmetric
matrices is a vector space without any computer or other pieces of
paper, then they don't understand what a vector space is. I would
say the same thing in an abstract algebra course with the definition
of a group, or a ring, or a module over a ring, etc. If you cannot give
a definition that is logically equivalent to the one in the book, you
don't know the subject. You have to have such in order to show a structure
is a group, or a ring, or a module, etc. You certainly need one in order
to prove anything about them.
--Dan Grubb
Daniel Grubb
>>Define "vector". You can't really, since you haven't properly defined
>>a vector space. Hint: axioms.
>I don't understand. A vector is any collection: (x1, ...., x^n) of anything.
>In math, the vectors are numbers.
This is wrong. A function can be a vector (usefully so, in fact).
A polynomial can be a vector, a sequence can be a vector. In fact,
*anything* can be a vector. All that it means to be a vector is to
be an element of a vector space (usually the specific vecvtor space
is understood from context).
--Dan Grubb
Daniel Grubb
>Right. That's my point. I *do* know the definition, but the phrasing always
>gets me. It's like this, probably, for a bunch of the main terms. Still, you
>say the number one problem is that students don't understand all the
>definitions. Is my understanding of "independence" then not good enough? Or
>is it?
Given what you claimed was a definition, I can say that your understanding
is not good enough. Not even close.
--Dan Grubb
George Cox
>
> In article <408EC9A3...@SPAMbtinternet.com.invalid>,
> George Cox <george_...@SPAMbtinternet.com.invalid> wrote:
>
> [Vector space.]
>
> >What is the precise
> >definition? If the precise definition has words like "abelian group" in
> >it, then what is the precise meaning of them? And so on.
>
> Yeah, sure. A vector space is a module over a field. That's absolutely
> true and probably given that way in Bourbaki somewhere, but even in
> selective institutions I can't believe students learn about modules
> before vector spaces. Is there really a source anywhere that pretends to
> _introduce_ vector spaces in terms of abelian groups?
I was taught (33 years ago in the University of Birmingham, UK) that a
vector space is
(1) a set of elements (the vectors) which form an abelian group under
addition;
(2) etc....
The phrase "commutative group" or "additive group" might have been used
instead of "abelian group"--you'll forgive me if I can't quite remember!
The course was the first algebra course in the first year of a BA.
How do you/US universities/today's commonly used texts define vector
space?
>
> Hmm, I suppose some here would argue that it messes up students'
> understanding of modules if they are first tainted by facts learned
> for vector spaces, which then have to be unlearned in the general case.
>
> dave
--
G.C.
Note ANTI, SPAM and invalid to be removed if you're e-mailing me.
George Cox
>
> In article <408EC9A3...@SPAMbtinternet.com.invalid>,
> George Cox <george_...@SPAMbtinternet.com.invalid> wrote:
>
> [Vector space.]
>
> >What is the precise
> >definition? If the precise definition has words like "abelian group" in
> >it, then what is the precise meaning of them? And so on.
All I meant was that if a wibbly is defined as a scrotted nup, one
doesn't _understand_ the definition of wibbly unless one understands
scrotted and nup.
>
> Yeah, sure. A vector space is a module over a field. That's absolutely
> true and probably given that way in Bourbaki somewhere, but even in
> selective institutions I can't believe students learn about modules
> before vector spaces. Is there really a source anywhere that pretends to
> _introduce_ vector spaces in terms of abelian groups?
>
> Hmm, I suppose some here would argue that it messes up students'
> understanding of modules if they are first tainted by facts learned
> for vector spaces, which then have to be unlearned in the general case.
>
> dave
Acid Pooh
> But, I've
> definitely always had some trouble doing proofs, even since sophomore
> year of high school, when I first encountered them in geometry. Do
> you think I should just drop the math major, or is a math major
> supposed to be a challenge for most people?
The way I see it, every major should be a challenge for most people.
>
> > You need to learn a method to go over the material quickly and
> > thoroughly. You need to learn how to come up with intuitive
> > representations of the concepts involved to guide your thinking. You
> > need to learn how to apply what you know to problem solving.
> >
> > So, what I suggest is that you stick it out. Read the definitions
> > until your head hurts.
>
> I've been doing this. It's starting to stick, finally.
>
> Try to figure out exactly what the quantifiers
> > are telling you. Do problems until your head hurts more and you dream
> > about mathematical symbols. Don't let a problem intimidate you.
>
> I'd love to do this, but I can't necessarily find many problems about
> the type of stuff we're supposed to be studying. It really annoys me
> when a textbook that doesn't have a lot of exercises doesn't include
> ALL answers in the back of the book. The text book seems to have just
> too few examples that it's not quite possible to "get" any one type of
> problem without other sources, like Schaum's.
What do you need answers for? Once you know the definitions and
theorems, logical implication is all you need to check the validity of
a proof. Granted, finding a proof is often very difficult. (OK, if
you end up having a doubt, have a friend look over it--or ask about it
here). But unless you're doing very computational problems, answer
keys are a red herring.
>
> >
> > If this sounds appealling, you probably have what it takes. If not, I
> > would suggest another major--it's not going to get any easier after
> > linear algebra.
>
> It sounds appealing, I'd say. Will it get any *harder* after linear
> algebra?
>
Short Answer: Yes. Long Answer: Absolutely, yes. :)
Seriously, your mathematical maturity grows in time (and with
practice). This is nothing you should worry about unless you don't
want to commit for some reason. The courses I'm in now are orders of
magnitude harder than undergraduate linear algebra, but my previous
courses have prepared me for these.
'cid 'ooh
Hiu Chung Law
> Grey Knight:
>>If you calculate det(<>-l*<>) for an n-by-n matrix <>, you
>>will get a degree-n polynomial in l; generally you'll be expected to
>>deal with 2-by-2 matrices, so this polynomial is just a quadratic. The
>>solutions of this polynomial are your eigenvalues.
> Thanks, Grey Knight. I've got that part down.
>>The eigenvectors
>>can then be found by solving the problem <> <x_i> = l_i <x_i>; that
>>is (<> - l_i<>)<x_i> = <0>
> This is the part I don't get. I'm sorry but your notation loses me. I see the
> "<>", and I think "dot product".
> We find the eigenvalues by solving matrix A....det (A-[lambda]*I ), where I is
> the identity matrix.
What Gary is saying is this.... Suppose that A is a 2 by 2 matrix
(a b)
(c d)
and then lambda is the eigenvalue. You should then solve for x1 and x2 in
(a-lambda b ) (x1) = (0)
(c d-lambda) (x2) (0)
Although there are two equations here, one of them is redundant. So you
can just solve either one... Since you have one equation, two unknowns,
there can be infinite solutions on x1 and x2. For simplicity, just pick
x1=1 and try to solve x2. If you can solve it, then you are done,
after you normalize the eigenvectors to norm one. If you get contradiction
by assuming x1=1, then x1 should be 0.
>>Two matrices are similar iff
>> <> = <<X>> <> <<X'>>
> OK. So I = X*I*X'. So, X' is any matrix that when multiplied by X gives the
> identity element as well?
Actually... I think there is a mistake here.
A n by n matrix X is similar to a n by n Y if you can find a n by n matrix T
such that
X = T * Y * inv(T)
where inv(T) denotes the inverse of T, inv(T) * T = I
Inverse and transpose are two different concepts
If A is
1 2
3 4
then A' (A transpose) is
1 3
2 4
and A inverse is
( 4 -3 )
( -2 1 ) / (-2)
> What's the difference between A transpose and A inverse? A specific example(s)
> would help. I'd love to test my skills.
Good luck to your exam.
Hiu Chung Law
> George:
>>I was taught (33 years ago in the University of Birmingham, UK) that a
>>vector space is
>>(1) a set of elements (the vectors) which form an abelian group under
>>addition;
>>(2) etc....
>>The phrase "commutative group" or "additive group" might have been used
>>instead of "abelian group"--you'll forgive me if I can't quite remember!
>>
> I think you just made me remember what an abelian group is. It's closed under
> its operation, it's associative, commutative, it's got an inverse, and it's got
> the identity element in it. There may be some other condition, since aren't
> all the above needed for any group? I remember something about an abelian
> group forming a table with a diagonal, or something....
> So, basically, "abelian group" is just an easier way of saying "vector space".
> Rather than listing all 8 conditions. But, that assumes the student *knows*
> what an abelian group is.
Well..... there is a big difference....
As you have already listed the 8 conditions, they are required for the
definition of a vector space. However, some of the 8 conditions can be
violated by an abelian group. For example, scalar multiplication is
undefined for an abelian group in general.
It will help you to appreciate their difference if you compare the exact
defintions of these two terms....
>>How do you/US universities/today's commonly used texts define vector
>>space?
> The 8 conditions.
rus...@mdli.com
wrote:
>In article <c98b1ba0.04042...@posting.google.com>,
>DE781 <de...@aol.com> wrote:
>
>>I already bought the text, the Cliff's
>>Notes guide to linear algebra, a 2003-version of the Schaum's outline,
>>and I even have an old 1968 version of Schaum's that my grandmother
>>used when she majored in math.
>
>This is not your grandmother's Linear Algebra course!
>(I've always wanted to say something like that.)
Ha, yes. But grandma's book is perfectly OK; that's the edition I was
praising in my earlier post. I see at Amazon that some reviewers are
complaining about errata in the new book; that surprised me because I
had found my 1968 edition remarkably error free. (At least to *my*
eye.) One reviewer cites a very minor misprint on p. 6, which can be
seen in the sample pages -- but that misprint is *only* in the newer
edition, at a place where some perfectly OK notation was replaced by
something different but really no better, and a mistake crept in. I
was sorry to see that; perhaps older is better in this case. (Though
not much of an issue, I suspect.)
To DE781 (who should be studying, not reading this) I apologize for
not addressing his specific concerns about this book; for some reason
I skimmed over that part of his post. Well, if he needs more
explanation than is in Schaum, that's easily available elsewhere (such
as his textbook). Personally I found the explanations adequate --
indeed superior to those in many textbooks -- but the strong point of
Schaum is really more in the area of confusion elimination; it *has*
to be sparse to achieve that end, IMHO.
Dave Seaman
> George:
>>I was taught (33 years ago in the University of Birmingham, UK) that a
>>vector space is
>>(1) a set of elements (the vectors) which form an abelian group under
>>addition;
>>(2) etc....
>>The phrase "commutative group" or "additive group" might have been used
>>instead of "abelian group"--you'll forgive me if I can't quite remember!
>>
> I think you just made me remember what an abelian group is. It's closed under
> its operation, it's associative, commutative, it's got an inverse, and it's got
> the identity element in it. There may be some other condition, since aren't
> all the above needed for any group?
All except the "commutative" part. If the group operation happens to be
commutative, then the group is abelian.
>I remember something about an abelian
> group forming a table with a diagonal, or something....
Nope. "Abelian group" simply means "commutative group", and nothing
more.
> So, basically, "abelian group" is just an easier way of saying "vector space".
In approximately the same sense that "steering wheel" is just an easier
way of saying "automobile". You have omitted two of the three things
needed in order to have a vector space:
(1) A collection of vectors V, which form an abelian group with
respect to vector addition (this is the only part you got right),
(2) A scalar field F,
(3) An operation called scalar multiplication, which binds V and F
together. The operation is associative and satisfies the
distributive laws.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
Dave Rusin
DE781 <de...@aol.com> wrote:
>>(a) Prove that the set M of all n by n matrices is a vector space (using
>> familiar matrix addition and scalar multiplication.)
>
>Let set M be all nXn matrices:
[Stick to fewer than 80 columns. What you displayed was probably supposed
to be a "picture" of three matrices, A B and C.]
Niggling little point: "Let ... be all nXn matrices" is a bit wrong.
The thing you're defining isn't going to be a matrix, and I don't see how
it can be all matrices since there are different matrices and a thing
can't be both A and B if A and B are different. What you mean to say is
"Let ... be the SET of all nXn matrices."
Did you know that at this level, mathematics is 90% grammar? Diagram
your sentence: you have a singular subject and so you need a singular
subject-complement; "the set" works, but "all ..." does not.
>Let s, t be scalars, elements of R. And let the above 3 matrices be called A,
>B, C, respectively.
>
>I'd show that A + B =
[deleted]
>also exists in set M.
OK.
>Then show A + B (above) = B + A, which is true because (a11 + b11) = (b11 +
>a11), same for all (aij + bij) = (bij + aij), since addition of scalars is
>commutative.
Good. (By the way you can indicate subscripts with an underscore: write
a_ij or a_{i,j} or something like that.)
>Then show (A+B) + C = A + (B+C) (associative).
Yes. No need to show me the proof; it's similar to the previous one.
>Then I'd multiply the zero matrix by A and show that it equals 0 matrix, which
>is also in M.
NO! Being a vector space makes no demand that you be able to multiply two
elements of M together in any way. Maybe you mean to show that multiplying
A by the _scalar_ 0 gives the zero matrix.
>Multiply I^n by A, to show that I*A = A.
NO again; the key thing is to show that multiplying by the NUMBER 1 returns
the matrix A. (In other words, you need an axiom to prevent silly things
from slipping in under the radar as a "vector space". Many theorems would
fail if you didn't insist that 1*A = A because a person could say, "Oh,
here's my vector space, and my definition of scalar multiplication is
that c*A means the zero matrix, no matter what c is". This is the only
axiom that prevents that.)
>multiply s(A) and show that:
>is also in M.
OK
>(st)(A) = (s(tA))
This has to be verified; you haven't done it yet, but yes, it's pretty
trivial too.
>And, there's the eight, right?
Um, you can't have copied the axioms right. There are different ways to
present vector spaces but there have to be axioms which combine scalar
multiplication with both additions (of numbers and of vectors). Something
like:
for all numbers c, d and all vectors v we have (c+d) v = (c v) + (d v)
and another one. (P.S. -- don't neglect the quantifiers "for all...";
many people do and I think it only adds to the confusion.)
You will probably need to copy your book's set of axioms to the newsgroup
since it looks like they're a little different from what I think is
most common. (I don't suppose it says something like, "We say that
(V, +, *, 0) is a vector space if ... ", does it?)
>>What is its dimension?
>
>N, I guess, since it has N rows.
NO! Find a basis and count how many elements are in that basis.
Of course, this will probably get us into a discussion of what a basis is...
>So, in homogenous form, there is a set of N
>equations each with N variables, right?
This "homogeneous form" stuff which you've mentioned before does not
really apply. You must be thinking of some specific applications of
vector spaces.
>>(b) Prove that the map f(x) = x^t is a linear transformation from M to M
Apologies; I did not make this notation clear. Putting a little "t" northeast
of the name of a matrix means to take its transpose. Maybe your book writes
x' for this, or uses some other notation.
>For any matrix A in set M (same A as above), f(x) = x^t maps every element aij
>in A to bij = (aij)^2 in matrix B.
Ack! Ptui! I hope first of all that you don't think that this is how you
square a matrix. Second, neither the matrix-squaring map f(x) = x^2,
nor the "Kronecker-square" map which you just described, is a linear
transformation! (New exercise: prove this!)
[wildly bogus proof deleted]
>>What is its kernel?
Try again for the transpose map.
Incidentally, if you apply the definition of the kernel to the (nonlinear)
map f(x) = x^2 you get non-trivial things in the "kernel" when n > 1.
>>(c) Compute the eigenvalues of f and find the eigenspaces.
>
>I don't know how this would work without actual numbers and an actual matrix.
>Since the example is an nXn matrix, we don't know how to calculate the
>determinant. We need det (A-lambda*I) to find the eigenvalues and
>eigenvectors.
NO! You yourself sort of gave the definition of eigenvectors in an
earlier post. You did not mention determinants, nor should you have!
I will give a hint: What is f o f ? (That is, what happens if you apply
f twice in a row?)
dave
Lee Rudolph
>>Let set M be all nXn matrices:
>
>[Stick to fewer than 80 columns.
Oh, all right, if you insist.
Let set M be all nX79 matrices.
Lee Rudolph
rus...@mdli.com
>Russell:
>
>>>It's a group of vectors that can be multiplied by any scalar and/or
>>>added together in any way,
>>
>>Not just any way -- there are precise conditions that the sum must
>>satisfy. Look for the axiomatic definition in your book(s). It's not
>>at all difficult to memorize. Do it.
>
>I already have memorized it. The space has to be closed under addition and
>scalar mult, and contain the zero vector. Therefore,
Not therefore -- some of the properties you list below are *in
addition to* those above.
it satisfies those 8
>properties:
>
>*v* + *w* = *x* (in the space)
>c*v* = c*v* (in the space)
>0*v* = 0
>1*v* = *v*
>
>*v* + *w* = *w* + *v*
>(*v* + *w*) + *x* = *v* + (*w* + *x*)
>*v* - *v* = 0
>c(s*v*) = (cs)(*v*)
Good. But you also need c(v+w) = cv + cw.
>
>>The algebra is the important thing, not the picture. And your current
>>understanding of the algebra is insufficient to keep you from getting
>>confused on the upcoming test.
>
>But, I did so well in algebra in 7th-9th grades. How much different can this
>be?
By that I meant only that you need to focus more on the equations,
symbols, etc. and worry less about the geometrical interpretations;
you seem to have an insufficiently abstract notion of what a vector
is. I see you've made a good start by listing axioms instead of
talking vaguely about planes and solids.
Regarding why this is harder, who really knows. But as (I think)
Prof. Ullrich said, this is typically the course where students who
are "good at math" first start having real difficulty. The habits
that you learn in passing this course will stand you in good stead in
the future. Even though (as Acid Pooh said) it won't get easier, one
of the things you are struggling with now is attitude, and once you
win that battle, you won't have as much difficulty with *that* as you
are having now. Everyone has to get over the hurdle sooner or later;
it's called mathematical maturity.
> I've got the definitions well enough: I got a B- on the first exam. It's
>the *concepts* since image/kernel/basis that have confused me, like I said.
Well, what folks are telling you here is that if you find yourself
confused, and you can't quickly write out a precise definition of the
thing you are working on, then *that* is the place to start getting
yourself unconfused. You flunked the initial test here in sci.math,
but you are starting to make a good comeback.
>
>>>I guess I could also really use some help with understanding how a
>>>mapping gets converted into a matrix, and then how to solve it.
>>
>>I like the term "linear transformation" and I think you should use it
>>too;
>
>OK. See? I never knew what that term meant. Now I do.
>
>>Do you know
>>what restrictions I'm talking about? f(a+b) = f(a)+f(b) and
>>f(ca) = cf(a) of course. That, by definition, is what makes the
>>mapping *linear*.
>
>Right. And also the 0 vector.
No, I think you aren't getting it here. The zero vector is a member
of a *linear space* -- i.e. a vector space. It has nothing really to
do with a *linear transformation*, which is a mapping between spaces.
The word "linear" appears in both phrases, and for similar reasons,
but they are not the same.
>
>>Try working it out for the 90-degree rotation I mentioned
>>above, and then try your matrix out on some 2D column vectors to see
>>if they really do turn 90 degrees when you multiply by the matrix.
>
>If something is rotated 90 degrees, the first point, cosine, goes from 1 to 0.
>And sine goes from 0 to 1. I still don't really get how to represent that in a
>matrix.
First of all, sine and cosine aren't points; instead, they are (or
rather they give) the two components of a *single* point (aka vector)
on the unit circle. They are indeed relevant to this problem, but
let's not think in terms of sines and cosines just yet.
Think basis vectors. What is the basis we are using here?
>
>Theta(pi/2) of (0,1) becomes (1,0).
I think you've rotated in the wrong direction; positive pi/2 is a
counterclockwise rotation. The vector (0,1) transforms to (-1,0);
that is, a unit y-vector rotates counterclockwise to a unit vector in
the -x direction. Since this y-vector was our second basis vector, we
now have the *second* column of our matrix. But we still need to find
the first column. Hint: how does a unit x-vector transform in a pi/2
(counterclockwise) rotation?
> Does that mean the matrix is just:
>
>/ 1\
>| 0|
>\ /
>2X1
Our vector space is R^2, so there are *two* basis vectors, and each of
them is a 2-tuple. So the matrix should be 2x2. Hopefully you have
worked the matrix out in your head by now, so I'll write it down:
0 -1
1 0
That is, the first column is a unit in the y direction, and the second
column is a unit in the -x direction.
We're done; but it never hurts to check our result. (And hopefully,
the following will give you some insight.) Let's see if this matrix
works for some arbitrary vector, say, (2,1). Express that vector as a
column vector, multiply by our matrix, and you get the answer (-1,2)
expressed as a column vector, right? Plot the two points (2,1) and
(-1,2) on some graph paper and draw in the lines from the origin; what
angle do you see? We have just demonstrated that multiplying by our
matrix has the effect of rotating our (arbitrary) vector by pi/2.
In other words, our matrix expresses the pi/2-rotation linear
transformation in the basis {(1,0),(0,1)}. A different basis would
give a different matrix for the same linear transformation. In fact
one of the things you may be asked to do is, given a matrix in one
basis, compute the matrix for the same linear transformation in a
different basis. I won't go into that here; it's in all of your books
and hopefully you have enough insight now to be able to understand
what they are talking about, when you read the relevant passages.
>
>?
>
>
>>(I am assuming you multiply with matrix on the left and column vector
>>on the right, as is done in Schaum's outline; hopefully that is how
>>your prof does it too,
>
>Yes.
Some books do it the other way; don't worry about that now, stick to
the convention used in your class.
rus...@mdli.com
>On 28 Apr 2004 22:23:59 GMT, de...@aol.com (DE781) wrote:
...
>>*v* + *w* = *x* (in the space)
>>c*v* = c*v* (in the space)
>>0*v* = 0
>>1*v* = *v*
>>
>>*v* + *w* = *w* + *v*
>>(*v* + *w*) + *x* = *v* + (*w* + *x*)
>>*v* - *v* = 0
>>c(s*v*) = (cs)(*v*)
>
>Good. But you also need c(v+w) = cv + cw.
Oops, and you need one more besides that; it's one you've already
written elsewhere.
I seem to be dualing with Dave Rusin in my advice. (That's a joke
btw; in fact I don't disagree with him at all.)