Chapter 8.8, Problem 23E

To determine

To describe: the resulting image if a cube is transformed first by matrix $U$ and then by matrix $T$ .

Answer to Problem 23E

The transformation will first reflect the figure over all the three coordinate planes and then second transformation will stretch the dimensions along the $y\text{- and }z\text{-}$ axes and also skews it along the $xy$ -plane such that first row of matrix $T$ will change the $x$ -coordinate to $\left(x+2z\right)$ .

Explanation of Solution

Given information:

  $\begin{array}{ccc}U=\left[\begin{array}{ccc}\begin{array}{l}-1\\ 0\\ 0\end{array}& \begin{array}{l}0\\ -1\\ 0\end{array}& \begin{array}{l}0\\ 0\\ -1\end{array}\end{array}\right]& \text{and}& T=\left[\begin{array}{ccc}\begin{array}{l}1\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 2\\ 0\end{array}& \begin{array}{l}2\\ 0\\ 2\end{array}\end{array}\right]\end{array}$

Calculation:

If a cube is transformed by matrix $U$ , then the transformation will reflect the figure over all the three coordinate planes. The transformation by matrix $T$ will stretch the dimensions along the $y\text{- and }z\text{-}$ axes and also skews it along the $xy$ -plane. So the first row of matrix $T$ will change the $x$ -coordinate to $\left(x+2z\right)$ .