Page 258 - 35Linear Algebra

258 Orthonormal Bases and Complements

This is an orthonormal basis since,for example:

√ √ Z 1
D 1 E 1
2 3 x − , 1 = 2 3 x − dx = 0 ,
2 0 2
and
Z 1
1 1 1 2 1 1
D E 2
x − , x − = x − dx = = √ .
2 2 0 2 12 2 3
An arbitrary vector v = a + bx is given in terms of the orthonormal basis O by
b ! b !
b 1 √ 1 a + 2 a + 2
v = (a + ).1 + b x − = 1, 2 3 x − = .
b
b
2 2 2 √ √
2 3 2 3 O
0
0
Hence we can predict the inner product of a + bx and a + b x using the dot product:
 
b ! 0 b 0
a + a + 0 0
2 2 b 0 b bb 0 1 0 0 1 0
·   = a + a + + = aa + (ab + a b) + bb .
b
b
√ √ 0 2 2 12 2 3
2 3 2 3
Indeed
1 1 1
Z
0
0
0
0
0
0
0
0
ha + bx, a + b xi = (a + bx)(a + b x)dx = aa + (ab + a b) + bb .
0 2 3
14.3 Relating Orthonormal Bases
Suppose T = {u 1 , . . . , u n } and R = {w 1 , . . . , w n } are two orthonormal bases
n
for R . Then
w 1 = (w 1 u 1 )u 1 + · · · + (w 1 u n )u n
. . .

w n = (w n u 1 )u 1 + · · · + (w n u n )u n
X
⇒ w i = u j (u j w i )
j

Thus the matrix for the change of basis from T to R is given by

j
P = (p ) = (u j w i ).
i
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