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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