Exercise 15 — Linear Algebra

Consider a linear transformation of $\mathbb{R}^{2}$ determined by these two.

\begin{pmatrix}1\\ 1\end{pmatrix}\mapsto\begin{pmatrix}2\\ 0\end{pmatrix}\qquad\begin{pmatrix}1\\ 0\end{pmatrix}\mapsto\begin{pmatrix}-1\\ 0\end{pmatrix}

Represent this transformation with respect to the standard bases.
Where does the transformation send this vector?
$\begin{pmatrix}0\\ 5\end{pmatrix}$
Represent this transformation with respect to these bases.
$B=\langle\begin{pmatrix}1\\ -1\end{pmatrix},\begin{pmatrix}1\\ 1\end{pmatrix}\rangle\qquad D=\langle\begin{pmatrix}2\\ 2\end{pmatrix},\begin{pmatrix}-1\\ 1\end{pmatrix}\rangle$
Using $B$ from the prior item, represent the transformation with respect to $B,B$ .

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