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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