Exercise2

This exercise concerns rotations and reflections in \(\real^2\text{.}\)

  1. Suppose that \(A\) is the matrix that performs a counterclockwise rotation in \(\real^2\text{.}\) Draw a typical picture of the vectors that form the columns of \(A\) and use the geometric definition of the determinant to determine \(\det A\text{.}\)

  2. Suppose that \(B\) is the matrix that performs a reflection in a line passing through the origin. Draw a typical picture of the columns of \(A\) and use the geometric definition of the determinant to determine \(\det A\text{.}\)

  3. As we saw in <<sec-transforms-geom>>, the matrices have the form

    \begin{equation*} A = \left[\begin{array}{rr} \cos \theta \amp -\sin\theta \\ \sin \theta \amp \cos \theta \\ \end{array}\right], \qquad B = \left[\begin{array}{rr} \cos(2\theta) \amp \sin(2\theta) \\ \sin(2\theta) \amp -\cos(2\theta) \\ \end{array}\right] \text{.} \end{equation*}

    Compute the determinants of \(A\) and \(B\) and verify that they agree with what you found in the earlier parts of this exercise.

in-context