Exercise 18

Find a matrix having this effect.

That is, find a $M$ that left-multiplies the starting vector to yield the ending vector. Is there a matrix having these two effects?

• (a)  $\begin{pmatrix}1\\ 3\end{pmatrix}\mapsto\begin{pmatrix}1\\ 1\end{pmatrix}\quad\begin{pmatrix}2\\ -1\end{pmatrix}\mapsto\begin{pmatrix}-1\\ -1\end{pmatrix}$     (b)  $\begin{pmatrix}1\\ 3\end{pmatrix}\mapsto\begin{pmatrix}1\\ 1\end{pmatrix}\quad\begin{pmatrix}2\\ 6\end{pmatrix}\mapsto\begin{pmatrix}-1\\ -1\end{pmatrix}$

Give a necessary and sufficient condition for there to be a matrix such that $\vec{v}_{1}\mapsto\vec{w}_{1}$ and $\vec{v}_{2}\mapsto\vec{w}_{2}$.