Exercise 3.2.5. If is a vector in then what is the angle between and the coordinate axes? What is the matrix that projects vectors in onto ?

Answer: Consider the coordinate axis and the unit vector lying along that axis, whose entry is 1 and whose other entries are zero. If is the angle between and then we have . We have and so that . We also have since the entries of and are both 1 and all other entries of are zero.

We thus have so that .

The matrix that projects vectors in onto is

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, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.