Suppose \displaystyle{ A = {\left[\begin{array}{cc}
11 &12\cr
-6 &-7
\end{array}\right]} } and f(\mathbf{x}) = A\mathbf{x} .

- 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>} } - The linear transformation
f acts like multiplication byin the subspace \mathrm{span}\lbrace {\left<2,-1\right>} \rbrace . - Is the vector
{\left<2,-1\right>} an eigenvector forA ?choose yes no cannot be determined - Is the vector
{\left<4,-2\right>} an eigenvector forA ?choose yes no cannot be determined - Are the vectors
{\left<2,-1\right>} and{\left<4,-2\right>} in the same eigenspace?choose yes no cannot be determined - If the vector
{\left<-1,k\right>} is an eigenvector forA with eigenvalue-1 , thenk = - 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>. - Is the vector
{\left<-3,3\right>} an eigenvector forf ?choose yes no cannot be determined

You can earn partial credit on this problem.