> On page 99, it says that "we want to think of ||z||^2 as the inner
> product of z with itself," but his formula for it looks like the inner
> product of z with the conjugate of z. I wanted to know what I am not
> seeing.
For complex n-space, the usual "dot product" formula for the inner
product is modified: you through in a complex conjugation. This has
the result that distance can be expressed in terms of the inner
product, as you say. In many linear algebra books, F can really be
any old field, and one ends up with a strange definition -- namely,
you do one thing in all cases *except* the case where F=C, when you do
something special. In our book, there are only two F's, namely R and
C. The definition is to one one thing in case of R and another, only
slightly different, thing in the case of C. This seems nicer because
C represents 50% of all fields.
> On page 104, I am having trouble understanding the proof of the
> Cauchy-
> Schwarz Inequality where Axler looks at what ||u||^2 is equal to based
> on the orthogonal decomposition. How does he get to the second
> equality (this equality is right before 6.8)?
He's using the principle that the square of the length of a.v (where a
is a scalar and v is a vector) is a^2 times the square of the length
of v. The scalar "a" here is a fraction whose denominator happens to
be the square of the length of v, so there is some cancellation. Note
that the denominator of a^2 is the fourth power of the length of v, so
you're still left with the square of the length of v in the
denominator after cancellation.
This proof is a tad different from the very common proof that exploits
the fact that a quadratic polynomial (over R) that's never 0 must have
a negative discriminant. Maybe I'll give the more common proof in
class, so that you'll have two proofs.
> On page 105, in the proof of the Triangle Inequality, how do I know
> that 2|<u,v>| is greater than or equal to 2Re<u,v>?
The inner product (u,v) is some number a+bi. Its absolute value is
the square root of a^2 + b^2. He's saying that a is less than or
equal to this square root. If a is negative, this is obvious because
the square root is always 0 or a positive number. If a is positive or
zero, then you say that the square root of a^2+b^2 is at least as big
as the square root of a^2, which is a. Geometrically, we're
confronting the statement that the hypotenuse of the right triangle
with sides |a| and |b| is bigger than either of the other sides. This
is kind of the triangle inequality for numbers, and it's being
parlayed into the triangle inequality for vectors. Good work if you
can get it.
Thanks for your questions!
Ken R
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