Exercise 3.4.15. Given the matrix

find the orthonormal vectors and that span the column space of . Next find the vector that completes the orthonormal set, and describe the subspace of of which is an element. Finally, for find the least squares solution to .

Answer: With and as the columns of , we first choose . We then have

Now that we have calculated the orthogonal vectors and we can normalize them to create the orthonormal vectors and . We have

so that

Since and span the column space of , and the orthonormal vectors and are linear combinations of and , and also span the column space of .

Next, we calculate . We can do this by orthogonalizing any vector that is linearly independent of and . For ease of calculation we start with . We then have

To normalize we divide by

so that

Of the four fundamental subspaces of , the left nullspace is orthogonal to the column space . Since and span the column space and is orthogonal to and , must be an element of the left nullspace .

Finally, to find the least squares solution to where , we factor and take advantage of the fact that .

The matrix is simply the 3 by 2 matrix with columns and :

The upper triangular matrix is a 2 by 2 matrix with

On the right side of the equation we have

so that the entire system is then

From the second equation we have or . Substituting into the first equation we have so that or .

The least squares solution to with is therefore .

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fifth Edition and the accompanying free online course, and Dr Strang’s other books.