Page 503 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

One could solve this problem by first using the Gram–Schmidt process to convert into an orthonormal basis and

then applying the method used in Example 6 of Section 6.3. However, the following method is more efficient.

The subspace W of spanned by , , and is the column space of the matrix

Thus, if u is expressed as a column vector, we can find the orthogonal projection of u on W by finding a least squares

solution of the system and then calculating from the least squares solution. The computations are as

follows: The system is

so

The normal system in this case is

Solving this system yields

as the least squares solution of (verify), so

or, in horizontal notation (which is consistent with the original phrasing of the problem), .

In Section 4.2 we discussed some basic orthogonal projection operators on and (Tables 4 and 5). The concept of an
orthogonal projection operator can be extended to higher-dimensional Euclidean spaces as follows.

DEFINITION