Exercise 6

Consider the linear transformation $t$ of $\mathbb{R}^{3}$ represented with respect to the standard bases by this matrix.

• Compute the determinant of the matrix. Does the transformation preserve orientation or reverse it?

• Find the size of the box defined by these vectors. What is its orientation?

  $$\begin{pmatrix}1\\ -1\\ 2\end{pmatrix}\quad\begin{pmatrix}2\\ 0\\ -1\end{pmatrix}\quad\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}$$

• Find the images under $t$ of the vectors in the prior item and find the size of the box that they define. What is the orientation?