Review exercise 1.29. Find 2 by 2 matrices that will

(a) reverse the direction of a vector

(b) project a vector onto the axis

(c) rotate a vector counter-clockwise through 90 degrees

(d) reflect a vector about the line that is 45 degrees above the axis

Answer: (a) We want a matrix such that . We have where is the 2 by 2 identity matrix, and thus . So we have if

(b) We want a matrix such that where . By experiment we see that the following matrix will work

(c) We want a matrix such that where is a vector rotated 90 degrees counter-clockwise from . Such a matrix would send, e.g., the vector to and in general would send the vector to . By experiment we see that the matrix

will do this:

(d) We want a matrix such that where is a vector reflected about the line . Such a matrix would send, e.g., the vector to and in general would send the vector to . By experiment we see that the matrix

will do this:

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.