Page 248 - A Course in Linear Algebra with Applications

232 Chapter Seven: Orthogonality in Vector Spaces

Example 7.3.5
Find an orthonormal basis for the column space S of the ma-
trix
1 1 2'
1 2 3
1 2 1
.1 1 6.
In the first place the columns X\, X 2, X 3 of the matrix
are linearly independent and so constitute a basis of S. We
shall apply the Gram-Schmidt process to this basis to produce
an orthonormal basis {Yi, Y 2, Y 3} of S, following the steps in
the procedure.

Now compute the projection of X 2 on S\ =< Y\ >;

1
Px = <X 2, Y 1>Y l=3Y 1
1
\lJ

The next vector in the orthonormal basis is

1
y 2 = ( X 2 P l )
| | X 2 - P 1 l | " - 2 1
\-lJ

The projection of X 3 on S 2 =< Yi, Y 2 > is

/ 4 2 \
= < X 3, Yi > Y x+ < X 3, = 6Y1 - =
P 2 Y 2>Y 2 2Y 2
2
V4/