Exercise 3.4.16. Given the matrix whose columns are the following two vectors and [sic]:

factor as . If there are vectors with elements each, what are the dimensions of , , and ?

Answer: With and as the two 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

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

The matrix is the 2 by 2 matrix calculated as follows:

If there are vectors with elements each, since they form the columns of the shape of will be by . The matrix contains one orthonormal column for each column in , and its orthonormal columns have the same number of elements as the columns of , so is also by . Finally, is a square matrix with one column for each column of , so it is by .

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.