Suppose $\displaystyle{ A = {\left[\begin{array}{cc} 11 &12\cr -6 &-7 \end{array}\right]} }$ and $f(\mathbf{x}) = A\mathbf{x}$.
1. If possible, complete the following equations; otherwise, enter DNE.
$\displaystyle{ {\left[\begin{array}{cc} 11 &12\cr -6 &-7 \end{array}\right]} {\left[\begin{array}{c} 2\cr -1 \end{array}\right]} = }$ $\displaystyle{ {\left[\begin{array}{c} 2\cr -1 \end{array}\right]} }$
$\displaystyle{ f \left( {\left<4,-2\right>} \right) = }$ $\displaystyle{ {\left<4,-2\right>} }$
$\displaystyle{ f \left( {\left<4,-2\right>} \right) = }$ $\displaystyle{ {\left<-1,1\right>} }$
2. The linear transformation $f$ acts like multiplication by in the subspace $\mathrm{span}\lbrace {\left<2,-1\right>} \rbrace$.
3. Is the vector ${\left<2,-1\right>}$ an eigenvector for $A$? If so, what is its associated eigenvalue?
4. Is the vector ${\left<4,-2\right>}$ an eigenvector for $A$? If so, what is its associated eigenvalue?
5. Are the vectors ${\left<2,-1\right>}$ and ${\left<4,-2\right>}$ in the same eigenspace?
6. If the vector ${\left<-1,k\right>}$ is an eigenvector for $A$ with eigenvalue $-1$, then $k =$
7. The linear transformation $f$ acts like multiplication by $-1$ in the subspace $\mathrm{span}\lbrace$ $\rbrace$. Enter your answer as a coordinate vector of the form <1,2>.
8. Is the vector ${\left<-3,3\right>}$ an eigenvector for $f$? If so, what is its associated eigenvalue?

You can earn partial credit on this problem.