Exercise 2.6.5. Suppose we have two points and and a third point halfway between the first two points. Show that for any linear transformation represented by a matrix the (transformed) point is halfway between and .

Answer: If is halfway between and then we have . For example, if and then

We then have

But this implies that is halfway between and , which is what we were asked to show.

For example, suppose that

Then using our example points above we have

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.