Page 502 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

and find the orthogonal projection of b on the column space of A.

Solution

Here

Observe that A has linearly independent column vectors, so we know in advance that there is a unique least squares solution.
We have

so the normal system in this case is

Solving this system yields the least squares solution
From Formula 5, the orthogonal projection of b on the column space of A is

Remark The language used for least squares problems is somewhat misleading. A least squares solution of is not in
instead.
fact a solution of unless happens to be consistent; it is a solution of the related system

EXAMPLE 2 Orthogonal Projection on a Subspace

Find the orthogonal projection of the vector on the subspace of spanned by the vectors

Solution