Page 243 - A Course in Linear Algebra with Applications

7.3: Orthonormal Sets and the Gram-Schmidt Process 227

form an orthogonal set since the scalar product of any two of
them vanishes. To obtain an orthonormal set, simply multiply
each vector by the reciprocal of its length:

Example 7.3.2
The standard basis of R n consisting of the columns of the
identity matrix l n is an orthonormal set.
Example 7.3.3

The functions

2
\j2/irs\n. mx, m = 1, ,...
form an orthonormal subset of the inner product space
C[0, 7r]. For we observed in Examples 7.2.4 and 7.2.5 that
these vectors are mutually orthogonal and have norm 1.

A basic property of orthogonal subsets is that they are
always linearly independent.
Theorem 7.3.1
Let V be a real inner product space; then any orthogonal subset
of V consisting of non-zero vectors is linearly independent.

Proof
Suppose that the subset {vi,..., v n } is orthogonal, so that
< vi, Vj > = 0 if i / j . Assume that there is a linear relation
of the form ciVi + • • • + c n v n = 0. Then, on taking the inner
product of both sides with Vj, we get
n n
0 = ^ < QVi, Vj > = '^T jC i <Vi,Vj > = Cj < Vj,Vj >
i=l i=l
II 112
— c • v •