## 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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