Page 241 - A Course in Linear Algebra with Applications

7.2: Inner Product Spaces 225

Exercises 7.2
1. Which of the following are inner product spaces?
T
(a) R n where < X, Y > = -X Y;
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(b) R n where < X, Y > = 2X Y;
(c) C[0,1] where <f,g>= J^(f(x)+g{x))dx.
2. Consider the inner product space C[0, TT] where < , g > =
/
C f(x)g(x)dx; show that the functions 1/y/n, \J2pn cos mx,
m — 1,2,..., form a set of mutually orthogonal unit vectors.
3. Let to be a fixed, positive valued function in the vector
space C[a, b}. Show that if < , g > is defined to be
/
b
I f(x)w(x)g(x)dx,

then < > is an inner product on C[a, b]. [Here w is called a
weight function].
4. Which of the following are normed linear spaces?
(a) R 3 where ||X|| =xl + x%+ xj;
(b) R 3 where ||X|| = y/x\ + x\ - x\\
(c) R where ||X|| = the maximum of |xi|, \x2\-, \%?\-
5. Let V be a finite-dimensional real inner product space
with an ordered basis v i , . . . , v n . Define a^ to be < v^, Vj >.
If A = [dij] and u and w are any vectors of V, show that
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< u, w > = [u] A[w] where [ u] is the coordinate vector of u
with respect to the given ordered basis.
6. Prove that the matrix A in Exercise 5 has the following
properties:
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(a) X AX > 0 for all X;
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(b) X AX = 0 only if X = 0;
(c) A is symmetric.
Deduce that A must be non-singular.