Linear Algebra and Its Applications, Review Exercise 1.29

Review exercise 1.29. Find 2 by 2 matrices that will

(a) reverse the direction of a vector

(b) project a vector onto the x_2 axis

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

(d) reflect a vector about the line x_1 = x_2 that is 45 degrees above the x_1 axis

Answer: (a) We want a matrix A such that Ax = -x. We have Ix = x where I is the 2 by 2 identity matrix, and thus -Ix = -x. So we have Ax = -x if A = -I

A = \begin{bmatrix} -1&0 \\ 0&-1 \end{bmatrix}

(b) We want a matrix A such that Ax = b where b = (0, x_2). By experiment we see that the following matrix will work

A = \begin{bmatrix} 0&0 \\ 0&1 \end{bmatrix}

Ax = \begin{bmatrix} 0&0 \\ 0&1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ x_2 \end{bmatrix}

(c) We want a matrix A such that Ax = b where b is a vector rotated 90 degrees counter-clockwise from x. Such a matrix would send, e.g., the vector (2, 1) to (-1, 2) and in general would send the vector x = (x_1, x_2) to (-x_2, x_1). By experiment we see that the matrix

A = \begin{bmatrix} 0&-1 \\ 1&0 \end{bmatrix}

will do this:

Ax = \begin{bmatrix} 0&-1 \\ 1&0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} -x_2 \\ x_1 \end{bmatrix}

(d) We want a matrix A such that Ax = b where b is a vector reflected about the line x_2 = x_1. Such a matrix would send, e.g., the vector (2, 1) to (1, 2) and in general would send the vector x = (x_1, x_2) to (x_2, x_1). By experiment we see that the matrix

A = \begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix}

will do this:

Ax = \begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} x_2 \\ x_1 \end{bmatrix}

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.

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