Suppose $A = \begin{bmatrix} 25 & -20 \\ 30 & -25 \end{bmatrix}$ and $f(x) = Ax$.
a. If possible, complete the following equations; otherwise, enter DNE.
$\begin{bmatrix} 25 & -20 \\ 30 & -25 \end{bmatrix} \begin{bmatrix} -1 \\ -1 \end{bmatrix} = \begin{bmatrix} \\ \end{bmatrix}$
$f((-3, -3)) = \Box (-3, -3)$
$f((-3, -3)) = \Box (-2, -3)$
b. The linear transformation $f$ acts like multiplication by $\Box$ in the subspace span$\{(-1, -1)\}$.
c. Is the vector $(-1, -1)$ an eigenvector for $A$? choose $\Box$ If so, what is its associated eigenvalue? $\Box$
d. Is the vector $(-3, -3)$ an eigenvector for $A$? choose $\Box$ If so, what is its associated eigenvalue? $\Box$
e. Are the vectors $(-1, -1)$ and $(-3, -3)$ in the same eigenspace? choose $\Box$
f. If the vector $(-2, k)$ is an eigenvector for $A$ with eigenvalue $-5$, then $k = \Box$
g. The linear transformation $f$ acts like multiplication by $-5$ in the subspace span$\{\Box\}$. Enter your answer as a coordinate vector of the form <1,2>.
h. Is the vector $(4, 6)$ an eigenvector for $f$? choose $\Box$ If so, what is its associated eigenvalue? $\Box$
