Exercise 3.2.16. a) Given the projection matrix projecting vectors onto the line through and two vectors and , show that the inner products of with and with are equal.

b) In general would the angles between and and and be equal to each other? If , , and what are the cosines of the two angles?

c) Show that the inner product of and is the same as the inner products of with and with and explain why this is. What is the angle between the vectors and ?

Answer: a) We have so that

and

The inner product of with is then

and the inner product of with is then

So the inner products of with and with are equal.

b) Since and are arbitrary vectors, in general they would not make the same angle with respect to . For example, consider the case when , , and .

If is the angle between and then . Similarly, if is the angle between and then

In this case we have

So and . The two cosines are different and thus and are not equal to each other.

c) As noted above we have and . Their inner product is then

But this is the same as the inner products and of with and with respectively.

This can be understood geometrically as follows: The vector can be thought of as consisting of two components and where is the projection of onto and is the projection of onto the vector that is orthogonal to . The inner product between and is then

But is simply . Also, since is a projection on and is a projection onto their inner product is zero since is orthogonal to . We therefore have .

Similarly the vector can be thought of as consisting of two components parallel to and orthogonal to , with the inner product between and then being

We thus have .

Since and are both projections onto the angle between them is zero.

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.