Page 239 - A Course in Linear Algebra with Applications

7.2: Inner Product Spaces 223

S is the subspace generated by a given vector u, then s is the
projection of v on u in the sense of 7.1.
Example 7.2.12
Find the projection of the vector X on the column space of
the matrix A where

X = 1 and A =

Let S denote the column space of A. Now the columns
of A are linearly independent, so they form a basis of S. We
have to find a vector Y in S such that X — Y is orthogonal to
1
both columns of A; for then X — Y will belong to S - and Y
will be the projection of X on S. Now Y must have the form

Y = x

for some scalars x and y. Then if A\ and A^ are the columns
1
of A, the conditions for X — Y to belong to S - are

< X-Y, A x > = (l-x-3y) + 2(l-2x + y) + {l-x-4y) = 0

and

< X-Y, A > = 3(l~x-3y)-(l~2x+y)+4{l-x-4:y) = 0.
2

These equations yield x = 74/131 and y — 16/131. The pro-
jection of X on the subspace S is therefore

122
1 f \

i6i
\138/