## Linear Algebra and Its Applications, Exercise 2.6.1

Exercise 2.6.1. Specify the 2 by 2 matrix that has the following effects:

1. Rotating all vectors 90 degrees.
2. Projecting the resulting vectors onto the $x$ axis.

Answer: From the discussion in the first part of section 2.6 we know that the matrix $\begin{bmatrix} 0&-1 \\ 1&0 \end{bmatrix}$

will rotate vectors through 90 degrees. From the same discussion we know that the matrix $\begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix}$

will project vectors onto the $x$ axis.

To accomplish both operations in succession we therefore need to multiply vectors by the first matrix and then by the second; this corresponds to multiplying vectors by the product matrix $\begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix} \begin{bmatrix} 0&-1 \\ 1&0 \end{bmatrix} = \begin{bmatrix} 0&-1 \\ 0&0 \end{bmatrix}$

For example, this matrix will take the vector $(1, 1)$ and transform it into the vector $(-1, 0)$ $\begin{bmatrix} 0&-1 \\ 0&0 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 0 \end{bmatrix}$

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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