Worksheet 24

1. Compute the volume of the four-dimensional parallelopiped spanned by the vectors \(\begin{pmatrix} 1 \\ -1 \\ -1 \\ 1 \end{pmatrix}, \ \begin{pmatrix} 0 \\ 1 \\ -1 \\ 1 \end{pmatrix}, \ \begin{pmatrix} 1 \\ 0 \\ 1 \\ 1 \end{pmatrix}, \ \begin{pmatrix} -1 \\ -1 \\ -1 \\ 0 \end{pmatrix}\)

2. Show that indeed the matrix \(R_\theta := \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}\) does rotate the plane counter-clockwise through an angle $\theta$ from $x$-axis. Show that $R_\theta$ is orthogonal.

3. Let $f : \mathbb{R}^2 \to \mathbb{R}^2$ be a differentiable function. How much does $f$ infinitesimally scale the volume (area) at a point $p$?