Page 302 - Matrix Analysis & Applied Linear Algebra

298 Chapter 5 Norms, Inner Products, and Orthogonality

Orthonormal Sets

B = {u 1 , u 2 ,..., u n } is called an orthonormal set whenever u i =1
for each i, and u i ⊥ u j for all i = j. In other words,

1 when i = j,
u i u j =
0 when i = j.
• Every orthonormal set is linearly independent. (5.4.2)
• Every orthonormal set of n vectors from an n-dimensional space V
is an orthonormal basis for V.

Proof. The second point follows from the first. To prove the first statement,
suppose B = {u 1 , u 2 ,..., u n } is orthonormal. If 0 = α 1 u 1 +α 2 u 2 +···+α n u n ,
use the properties of an inner product to write
0= u i 0 = u i α 1 u 1 + α 2 u 2 + ··· + α n u n
2
= α 1 u i u 1 + ··· + α i u i u i + ··· + α n u i u n = α i u i
for each i.
= α i
Example 5.4.4
!
1 1 −1

The set B = u 1 = −1 , u 2 = 1 , u 3 = −1 is a set of mutually
0 1 2
T
orthogonal vectors because u u j =0 for i = j, but B is not an orthonormal

i
set—each vector does not have unit length. However, it’s easy to convert an
orthogonal set (not containing a zero vector) into an orthonormal set by simply
√ √ √
normalizing each vector. Since u 1 = 2, u 2 = 3, and u 3 = 6, it
√ √ √
follows that B = u 1 / 2, u 2 / 3, u 3 / 6 is orthonormal.
The most common orthonormal basis is S = {e 1 , e 2 ,..., e n } , the stan-
n n 2 3
dard basis for and C , and, as illustrated below for and , these
orthonormal vectors are directed along the standard coordinate axes.
y z
e 2 e 3

e 1 e 2
x y
e 1
x