## Linear Algebra and Its Applications, Exercise 2.6.6

Exercise 2.6.6. Suppose we have a transformation matrix $A = \begin{bmatrix} 1&0 \\ 3&1 \end{bmatrix}$

This is a shearing transformation: it leaves unchanged any points $(x, y)$ for which $x = 0$ and thus leaves unchanged the entire $y$-axis. Describe how this transformation affects the $x$-axis, including the points $(1, 0)$, $(2, 0)$, and $(-1, 0)$.

Answer: For the point $(1, 0)$ we have $\begin{bmatrix} 1&0 \\ 3&1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$

For the point $(2, 0)$ we have $\begin{bmatrix} 1&0 \\ 3&1 \end{bmatrix} \begin{bmatrix} 2 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 6 \end{bmatrix}$

For the point $(-1, 0)$ we have $\begin{bmatrix} 1&0 \\ 3&1 \end{bmatrix} \begin{bmatrix} -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -1 \\ -3 \end{bmatrix}$

Based on these examples it appears that the matrix $A$ transforms the $x$-axis into a line through the origin with slope 3.

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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