Page 245 - A Course in Linear Algebra with Applications

7.3: Orthonormal Sets and the Gram-Schmidt Process 229

Another useful feature of orthonormal bases is that they
greatly simplify the procedure for calculating projections.

Theorem 7.3.3
Let V be an inner product space and let S be a subspace and
v a vector of V. Assume that {si,... , s m } is an orthonormal
basis of S. Then the projection of v on S is

m
^2<V, Si> Si.

1 = 1

Proof
s
) i
Put p = Y^Li < v s > i> a vector which quite clearly
belongs to S. Now < p, s^- > = < v, Sj >, so

< V - p , Sj > = < V, Sj > - < p , Sj >
= < V, Sj > — < V, Sj >
= 0.

Hence v — p is orthogonal to each basis element of S, which
rJ
shows that v — p belongs to 5 -. Since v = p + (v — p), and
the expression for v as the sum of an element of S and an
1
element of S - is unique, it follows that p is the projection of
v on S.

Example 7.3.4
The vectors

3
form an orthonormal basis of a subspace S of R ; find the
projection on S of the column vector X with entries 1,-1,1.