6. Problems

(not to be done unless assigned)

Problem C-1. Find a matrix $A$ for which the linear map $T _ A($x$) = A$x has $T _ A \left[\begin{array}{r}1;0\end{array}\right] = \left[\begin{array}{r}2;3\end{array}\right]$ and $T _ A \left[\begin{array}{r}0;1\end{array}\right] = \left[\begin{array}{r}1;4\end{array}\right]$.

Problem C-2. Find a matrix $A$ for which the linear map $T _ A($x$) = A$   x has $T _ A(\left[\begin{array}{r}2\\ 3\end{array}\right]) = \left[\begin{array}{r}5\\ 7\end{array}\right]$ and $T _ A(\left[\begin{array}{r}1\\ 4\end{array}\right]) = \left[\begin{array}{r}6\\ 3\end{array}\right]$.

(Method: Find a matrix $B$ for which the linear map $U($x$) = B$   x gives $U(\left[\begin{array}{r}1\\ 0\end{array}\right]) = \left[\begin{array}{r}2\\ 3\end{array}\right]$ and $U(\left[\begin{array}{r}0\\ 1\end{array}\right]) = \left[\begin{array}{r}1\\ 4\end{array}\right]$ and a matrix $C$ for which the linear map $V($x$) = C$   x gives $V(\left[\begin{array}{r}1\\ 0\end{array}\right]) = \left[\begin{array}{r}5\\ 7\end{array}\right]$ and $V(\left[\begin{array}{r}0\\ 1\end{array}\right]) = \left[\begin{array}{r}6\\ 3\end{array}\right]$. Then use $A = C B ^ {-1}$. Why does this work?)

Problem C-3. In R$^ 2$, consider the square with vertices $\left[\begin{array}{r}\pm 1\\ \pm 1\end{array}\right]$. Write down matrices for all linear maps that take this square to itself, including the identity matrix. (There are eight possibilities.)

Problem C-4. For each map on the handout with images of a house, write down the corresponding matrix and its determinant. (Count picture #1 as being the identity map.)

(Method: Look at the images of the standard basis vectors $\left[\begin{array}{r}0\\ 1\end{array}\right]$ and $\left[\begin{array}{r}1\\ 0\end{array}\right]$; these give the rows of the matrix. If it's not clear which standard basis vector goes to which image vector, then look at how the house and its image are lined up with respect to the standard basis vectors and their images.)