Exercise 6 — Linear Algebra

Consider the spaces $V=\mathcal{P}_{2}$ and $W=\mathcal{M}_{2\!\times\!2}$ , with these bases.

\displaystyle B=\langle 1,1+x,1+x^{2}\rangle\qquad D=\langle\begin{pmatrix}0&0\\ 0&1\end{pmatrix},\begin{pmatrix}0&0\\ 1&1\end{pmatrix},\begin{pmatrix}0&1\\ 1&1\end{pmatrix},\begin{pmatrix}1&1\\ 1&1\end{pmatrix}\rangle

\displaystyle\hat{B}=\langle 1,x,x^{2}\rangle\qquad\hat{D}=\langle\begin{pmatrix}-1&0\\ 0&0\end{pmatrix},\begin{pmatrix}0&-1\\ 0&0\end{pmatrix},\begin{pmatrix}0&0\\ 1&0\end{pmatrix},\begin{pmatrix}0&0\\ 0&1\end{pmatrix}\rangle

We will find $P$ and $Q$ to convert the representation of a map with respect to $B,D$ to one with respect to $\hat{B},\hat{D}$

AI Tutor