Page 249 - A Course in Linear Algebra with Applications

7.3: Orthonormal Sets and the Gram-Schmidt Process 233

The final vector in the orthonormal basis of S is therefore

Y*

Example 7.3.6
Find an orthonormal basis of the inner product space P3 (R)
where < f,g > is defined to be J_ 1 f(x)g(x)dx.
2
We begin with the standard basis {1, x, x } of Pa(R) and
then use the Gram-Schmidt process to construct an orthonor-
mal basis {/1, fa, fa}. Since ||1|| = y{J_ l x) = \/2, the first
member of the basis is

1 - 1 1 - 1

Next <x,f x> = f^(x/V2)dx = 0, so p x =< x, 1>f 1 = 0.
f
Hence

2
since ||x|| = y/(f_ 1 x dx) = w | .
2
Continuing the procedure, we find that < x , fi > =
2
2
2
x/2/3 and < x , f 2 > = 0. Hence p 2 = < x , /1 > i + < x ,
/
H > H — 1/3, and so the final vector of the orthonormal
basis is
U = - (x 2 --) = ^(x 2 - -)

Consequently the polynomials

1 3 2 1
^ . a n d * - " !
V2' V 2 2V2