Page 120 - Linear Algebra Done Right

Orthonormal Bases
Corollary:
Every orthonormal list of vectors is linearly inde-
6.16
pendent.
Proof: Suppose (e 1 ,...,e m ) is an orthonormal list of vectors in V 107
and a 1 ,...,a m ∈ F are such that
a 1 e 1 +· · ·+ a m e m = 0.
2
2
Then |a 1 | +· · ·+|a m | = 0 (by 6.15), which means that all the a j ’s
are 0, as desired.
An orthonormal basis of V is an orthonormal list of vectors in V
that is also a basis of V. For example, the standard basis is an ortho-
n
normal basis of F . Every orthonormal list of vectors in V with length
dim V is automatically an orthonormal basis of V (proof: by the pre-
vious corollary, any such list must be linearly independent; because it
has the right length, it must be a basis—see 2.17). To illustrate this
4
principle, consider the following list of four vectors in R :
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
( , , , ), ( , , − , − ), ( , − , − , ), (− , , − , ) .
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
The verification that this list is orthonormal is easy (do it!); because we
have an orthonormal list of length four in a four-dimensional vector
space, it must be an orthonormal basis.
In general, given a basis (e 1 ,...,e n ) of V and a vector v ∈ V,we
know that there is some choice of scalars a 1 ,...,a m such that
v = a 1 e 1 +· · ·+ a n e n ,

but finding the a j ’s can be difficult. The next theorem shows, however,
that this is easy for an orthonormal basis.

6.17 Theorem: Suppose (e 1 ,...,e n ) is an orthonormal basis of V. The importance of
Then orthonormal bases
stems mainly from this
6.18 v = v, e 1 e 1 + ··· + v, e n e n theorem.

and

2
2
6.19 v =| v, e 1 | +· · ·+| v, e n | 2
for every v ∈ V.