Exercise 19

Let $t\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ represented by $T$ with respect to ${\cal E}_{n},{\cal E}_{n}$.

• Find ${\rm Rep}_{B,B}(t)$ in this specific case.

  $$T=\begin{pmatrix}1&1\\ 3&-1\end{pmatrix}\qquad B=\langle\begin{pmatrix}1\\ 2\end{pmatrix},\begin{pmatrix}-1\\ -1\end{pmatrix}\rangle$$

• Describe ${\rm Rep}_{B,B}(t)$ in the general case where $B=\langle\vec{\beta}_{1},\ldots,\vec{\beta}_{n}\rangle$.