Kolman's Elementary Linear Algebra book
genericaudioperson
I have been studying Kolman's Linear Algebra book. I was hoping
someone with years of high-level mathematics experience and who knows
this book can offer some insights.
What struck me about the book is the insane amount of proofs and proof-
like things. I didn't count all the chapters, but it looks like it
maps out well beyond 1000 times it asks you to "prove" "show" or
"explain". Even a true false question can be interpreted as a proof,
because you will need to explain your work. I particularly dislike
when in the text portion of the book it says Proof: exercise. And
then makes the suffering student have to figure out the proof, with no
way to check the answer to see if they did the proof correctly.
Anything very important should have the completed worked solution.
The student should not be left guessing. But onto the main
observations:
It seems like this book is trying to accomplish two purposes: 1) show
you how linear algebra works 2) force you to prove mathematical
statements. Do you agree with this observation?
This brings me to another idea: you're not really supposed to do
every single 1000+ proof or proof-like question. You are supposed to
select enough of them to sharpen your proof skills. But doing all of
them would be a waste of time. I say this, because even if you are
super dedicated, you must at some point ask yourself at what point
should you work on proving *other* math concepts with your available
time, rather than proving trivial point #787 in the Kolman book.
There are only 168 hours in the week.
So in summary, I have developed these observations:
1) The Kolman book is forcing you to work on proofs besides learning
linear algebra
2) You're not really supposed to complete every proof-type question in
the book. You select a reasonable amount.
Any thoughts on this? Thanks.
quasi
At some point, you may reach a level of strength with respect to
proofs, which makes it possible, for many problems, for you to quickly
see a proof strategy which will surely work. In that case, as long as
you are truly sure, there's no need to actually carry out the proof.
quasi
Dave L. Renfro
http://groups.google.com/group/sci.math/msg/6a214d83dd4f56a2
> I have been studying Kolman's Linear Algebra book. I was
> hoping someone with years of high-level mathematics
> experience and who knows this book can offer some insights.
>
> What struck me about the book is the insane amount of
> proofs and proof-like things. I didn't count all the
> chapters, but it looks like it maps out well beyond
> 1000 times it asks you to "prove" "show" or "explain".
> Even a true false question can be interpreted as a proof,
> because you will need to explain your work. I particularly
> dislike when in the text portion of the book it says
> Proof: exercise. And then makes the suffering student
> have to figure out the proof, with no way to check the
> answer to see if they did the proof correctly.
One reason for thinking there's an overemphasis on
proof and proof-like things is that you probably haven't
been exposed to much mathematics prior to this. Calculus,
elementary differential equations, and below consists
almost entirely of learning various algorithmic procedures,
which is not really mathematics, anymore than English
grammar is the subject matter that one studies in college
English courses or names and dates are the subject matter
of college history courses. If anything, I think other
fields get to the meat of their subject much earlier
than math does.
Granted, linear algebra has a strong computational
component, but at some point in the study of math
one has to begin learning simple proof ideas and
techniques, and linear algebra has traditionally
been one of the places this is done. You can't
just wait until advanced undergraduate courses
in real analysis and abstract algebra, because
most people can't make the transition so quickly.
Indeed, the fact that you're having trouble with
beginning linear algebra level proofs only strengthens
this argument.
> Anything very important should have the completed worked
> solution. The student should not be left guessing. But
> onto the main observations:
>
> It seems like this book is trying to accomplish two
> purposes: 1) show you how linear algebra works 2) force
> you to prove mathematical statements. Do you agree with
> this observation?
Actually, I consider (1) and (2) to be saying about the
same thing, since that's what proofs are -- showing
someone how/why a result is true. I think the additional
task you mean is 3) teach you some algorithmic procedures
in linear algebra, such as solving linear systems and
how to describe the solution set, how to change coordinates
relative to one ordered basis to another ordered basis,
how to represent by a matrix a linear function from one
finite dimensional vector space with a specified ordered
basis to another finite dimensional vector space with a
specified ordered basis, etc.
> This brings me to another idea: you're not really supposed
> to do every single 1000+ proof or proof-like question.
> You are supposed to select enough of them to sharpen
> your proof skills. But doing all of them would be a
> waste of time. I say this, because even if you are
> super dedicated, you must at some point ask yourself
> at what point should you work on proving *other* math
> concepts with your available time, rather than proving
> trivial point #787 in the Kolman book. There are only
> 168 hours in the week.
There are very few textbooks in which the number of
problems is so small that you should do all of them,
or even most of them. The problems are designed for
use in varied classroom settings, according to the
needs of the college, the needs of the students, the
interests of the instructor, etc.
> So in summary, I have developed these observations:
> 1) The Kolman book is forcing you to work on proofs
> besides learning linear algebra
> 2) You're not really supposed to complete every
> proof-type question in the book. You select a
> reasonable amount.
>
> Any thoughts on this? Thanks.
If anything, the Kolman book, like virtually every other
linear algebra book for college Freshmen and Sophomores,
is actually designed in large part for those outside of
mathematics (chemists, engineers, biologists, economists,
physicists, etc.), and thus it has a very strong emphasis
on the algorithmic procedures of linear algebra. And
you're right about not having to complete every proof-type
question in the book, no more than you're supposed to
evaluate every one of the hundreds of partial fraction
and trig. substitution type integrals in a typical calculus
text.
If you're studying this on your own, I'd suggest also
getting a copy of the "Schaum's Outline" for linear algebra
and/or one of the many other problem books in linear
algebra that you can find in most every university bookstore.
Some of these may be pitched at a higher level than the
Kolman book (such as the "Schaum's Outline" for linear
algebra), but I think you'll still find them very useful
when studying by yourself. Another thing you can do is to
check out from a college/university library (if one is
within driving distance from you) several linear algebra
texts that are about the same level as Kolman's text,
and then for each topic you cover, look over what each
author deems important enough to NOT put in the exercises.
In the several cases when I've learned mathematical topics
on my own, I've found this to be a very useful method in
sorting out the important stuff from the busy work, at
least before I know enough about the subject to trust
myself. Another suggestion is to at least read through
all the proof-like questions to make sure you understand
what is being said, because if you don't, this could mean
that you need to review some earlier material again.
Plus, some of the exercises (especially as you go
further in math) tend to have fairly interesting things
in them, but they're in the exercises because the author
deemed the "interesting things" too tangential to the
main flow of the text (or perhaps they require some
background knowledge in an area that is otherwise not
intended to be a prerequisite for the text, although
the author will usually say so when this is the case)
to put the "interesting things" in the text material.
Dave L. Renfro