Page 244 - A Course in Linear Algebra with Applications

228 Chapter Seven: Orthogonality in Vector Spaces

since < v i: Vj > = 0 if i ^ j . Now ||vj|| ^ 0 since Vj is not
j
the zero vector; therefore Cj — 0 for all . It follows that the
Vj are linearly independent.
This result raises the possibility of an orthonormal basis,
and indeed we have already seen in Example 7.3.2 that the
standard basis of R n is orthonormal. While at present there
are no grounds for believing that such a basis always exists,
it is instructive to record at this stage some useful properties
of orthonormal bases.

Theorem 7.3.2
Suppose that {vi,...,v n } is an orthonormal basis of a real
inner product space V. If v is an arbitrary vector of V, then

n n
v — ^ < v, Vj > Vj and ||v|| 2 = ^ < v, Vj > 2 .
i = l i = l

Proof
c v
e
Let v = X)I=i i « t» the expression for v in terms of the
given basis. Forming the inner product of both sides with Vj,
we obtain
n n
C V
< V, Vj > = < ^2 * *> Vj > = J ^ Cj < Vj, Vj > = Cj
i=l i=l

since < Vj, Vj > = 0 if i ^ j and < Vj, v,- > = 1. Finally,
n n
C V
||v|| 2 = < v, v > = < ^ c i V i , Y; J J >
t = l 3=1
n n
= ^Yl CiC i <x ^j >,
i = i j = i

c
which reduces to Y^i=i j-