Page 232 - A Course in Linear Algebra with Applications

216 Chapter Seven: Orthogonality in Vector Spaces

Example 7.2.7
n
The Euclidean space R is a normed linear space if length is
taken as the norm. Thus

||X|| = VX^X = /xl+x% + .-- + xl

Example 7.2.8
The vector space C[a,b] becomes a normed linear space if ||/||
is defined to be
1 2
{f' f(xfdx) / .
J a
Example 7.2.9 (Matrix norms)
A different type of normed linear space arises if we consider
the vector space of all real m x n matrices and introduce a
norm on it as follows. If A = [ciij\m,ni define \\A\\ to be

m n
1/2
<E£4) -

On the face of it this is a reasonable measure of the "size" of
the matrix. But of course one has to show that this is really a
norm. A neat way to do this is as follows: put A equal to the
ran-column vector whose entries are the elements of A listed
by rows. The key point to note is that \\A\\ is just the length
of the vector A in R m n . It follows at once that || || is a norm
since we know that length is a norm.

Inner products on complex vector spaces
So far inner products have only been defined on real vec-
tor spaces. Now it has already been seen that there is a rea-
sonable concept of orthogonality in the complex vector space
n n
C , although it differs from orthogonality in R in that a dif-
ferent scalar product must be used. This suggests that if an