Exercise 7

For each space find the matrix changing a vector representation with respect to $B$ to one with respect to $D$.

• $V=\mathbb{R}^{3}$, $B={\cal E}_{3}$, $D=\langle\begin{pmatrix}1\\ 2\\ 3\end{pmatrix},\begin{pmatrix}1\\ 1\\ 1\end{pmatrix},\begin{pmatrix}0\\ 1\\ -1\end{pmatrix}\rangle$

• $V=\mathbb{R}^{3}$, $B=\langle\begin{pmatrix}1\\ 2\\ 3\end{pmatrix},\begin{pmatrix}1\\ 1\\ 1\end{pmatrix},\begin{pmatrix}0\\ 1\\ -1\end{pmatrix}\rangle$, $D={\cal E}_{3}$

• $V=\mathcal{P}_{2}$, $B=\langle x^{2},x^{2}+x,x^{2}+x+1\rangle$, $D=\langle 2,-x,x^{2}\rangle$