Page 260 - 35Linear Algebra

260 Orthonormal Bases and Complements

T
Definition A matrix P is orthogonal if P −1 = P .

Then to summarize,

Theorem 14.3.1. A change of basis matrix P relating two orthonormal bases
is an orthogonal matrix. I.e.,

T
P −1 = P .

Reading homework: problem 2

3
Example 134 Consider R with the ordered orthonormal basis
 2     1 
√ 0 √
6 3
√ √ √  .
 1   1   −1
6 2 3
S = (u 1 , u 2 , u 3 ) =   ,   , 
−1
√ √ 1 √ 1
6 2 3
Let E be the standard basis (e 1 , e 2 , e 3 ). Since we are changing from the standard
basis to a new basis, then the columns of the change of basis matrix are exactly the
new basis vectors. Then the change of basis matrix from E to S is given by
 
e 1 u 1 e 1 u 2 e 1 u 3
j
P = (P ) = (e j · u i ) =  e 2 u 1 e 2 u 2 e 2 u 3 
i
e 3 u 1 e 3 u 2 e 3 u 3
 2 1 
√ 0 √
6 3
 1 1 −1
= u 1 u 2 u 3 √ √ √  .
6 2 3
= 
−1
√ √ 1 √ 1
6 2 3
From our theorem, we observe that
 T 
u
1
u
P −1 = P T =  T 
2
u T
3
 2 1 −1 
√ √ √
6 6 6
0  .
 √ 1 √ 1 
2 2
= 
√ 1 −1 √ 1
√
3 3 3
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