Page 255 - A Course in Linear Algebra with Applications

7.3: Orthonormal Sets and the Gram-Schmidt Process 239

or
X
Q- =Q*.
T
Recall here that Q* = {Q) • A complex matrix Q with the
above property is said to be unitary. Thus unitary matri-
ces are the complex analogs of real orthogonal matrices. For
example, the matrix

( cos 9 isin9\
\isin9 cos9 J '

is unitary for all real values of 9; here of course i = \f—l.

Exercises 7.3

1. Show that the following vectors constitute an orthogonal
basis of R 3 :

'i)-G)-(4

2. Modify the basis in Exercise 1 to obtain an orthonormal
basis.
3. Find an orthonormal basis for the column space of the
matrix
0 1 1'
1 - 2 1
1 2 0

4. Find an orthonormal basis for the subspace of ^ ( R ) gen-
erated by the polynomials 1 — 6x and 1 — 6x 2 where < f,g >
= Jo f(x)g(x)dx.