Page 256 - 35Linear Algebra

256 Orthonormal Bases and Complements

in T, we get

1 i n
v u i = c u 1 u i + · · · + c u i u i + · · · + c u n u i
i
1
n
= c · 0 + · · · + c · 1 + · · · + c · 0
i
= c ,
⇒ c i = v u i
⇒ v = (v u 1 )u 1 + · · · + (v u n )u n
X
= (v u i )u i .
i

This proves the following theorem.

Theorem 14.2.1. For an orthonormal basis {u 1 , . . . , u n }, any vector v can
be expressed as
X
v = (v u i )u i .
i

Reading homework: problem 1

2
All orthonormal bases for R

14.2.1 Orthonormal Bases and Dot Products
n
To calculate lengths of, and angles between vectors in R we most commonly
use the dot product:

   1 
1
v w
n
n
1
1
.
.
 .  ·  .  := v w + · · · + v w .
 .   . 
v n w n
When dealing with more general vector spaces the dot product makes no
sense, and one must instead choose an appropriate inner product. By “ap-
propriate”, we mean an inner product well-suited to the problem one is try-
ing to solve. If the vector space V under study has an orthonormal basis
O = (u 1 , . . . , u n ) meaning
hu i , u j i = δ ij ,

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