## 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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