Page 119 - Linear Algebra Done Right

Chapter 6. Inner-Product Spaces
106
u
u + v
u − v
v v
u
The parallelogram equality
6.14 Parallelogram Equality: If u, v ∈ V, then

2
2
2
2
u + v + u − v = 2( u + v ).
Proof: Let u, v ∈ V. Then
2 2
u + v + u − v = u + v, u + v + u − v, u − v
2 2
= u + v + u, v + v, u
2 2
+ u + v −Pu, v −Pv, u
2
2
= 2( u + v ),
as desired.
Orthonormal Bases

A list of vectors is called orthonormal if the vectors in it are pair-
wise orthogonal and each vector has norm 1. In other words, a list
(e 1 ,...,e m ) of vectors in V is orthonormal if e j ,e k equals 0 when
j = k and equals 1 when j = k (for j, k = 1,...,m). For example, the
n
standard basis in F is orthonormal. Orthonormal lists are particularly
easy to work with, as illustrated by the next proposition.

6.15 Proposition: If (e 1 ,...,e m ) is an orthonormal list of vectors
in V, then
2 2 2
a 1 e 1 + ··· + a m e m =|a 1 | + ··· + |a m |
for all a 1 ,...,a m ∈ F.

Proof: Because each e j has norm 1, this follows easily from re-
peated applications of the Pythagorean theorem (6.3).

Now we have the following easy but important corollary.