Linear Algebra and Its Applications, Review Exercise 2.29

Review exercise 2.29. The following matrices

A_1 = \begin{bmatrix} 1&0 \\ 0&-1 \end{bmatrix} \qquad A_2 = \begin{bmatrix} 1&0 \\ 2&1 \end{bmatrix} \qquad A_3 = \begin{bmatrix} 0&1 \\ -1&0 \end{bmatrix}

represent linear transformations in the xy plane with e_1 = (1, 0) and e_2 = (0, 1) as a basis. Describe the effect of each transformation.

Answer: When the matrix A_1 is applied to the vector e_1 = (1, 0) we obtain

A_1v_1 = \begin{bmatrix} 1&0 \\ 0&-1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}

When the matrix A_1 is applied to the vector e_2 = (0, 1) we obtain

A_1v_1 = \begin{bmatrix} 1&0 \\ 0&-1 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ -1 \end{bmatrix}

The effect of the transformation represented by A_1 is to reflect all vectors through the x-axis.

When the matrix A_2 is applied to the vector e_1 = (1, 0) we obtain

A_2e_1 = \begin{bmatrix} 1&0 \\ 2&1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}

When the matrix A_2 is applied to the vector e_2 = (0, 1) we obtain

A_2e_2 = \begin{bmatrix} 1&0 \\ 2&1 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}

When the matrix A_2 is applied to the vector v_1 = (1, 2) we obtain

A_2v_1 = \begin{bmatrix} 1&0 \\ 2&1 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \end{bmatrix}

The transformation represented by A_2 is a shearing transformation that leaves all vectors on the y-axis unchanged but changes the y coordinates of all other vectors in proportion to their distance from the y-axis.

When the matrix A_3 is applied to the vector e_1 = (1, 0) we obtain

A_3e_1 = \begin{bmatrix} 0&1 \\ -1&0 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ -1 \end{bmatrix}

When the matrix A_2 is applied to the vector e_2 = (0, 1) we obtain

A_2e_2 = \begin{bmatrix} 0&1 \\ -1&0 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}

The effect of the transformation represented by A_3 is to rotate all vectors through an angle of -90 degrees. (Or, in other words, clockwise through 90 degrees.)

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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