Determine the eigenvalues of each linear transformation. Give brief explanations.
(Hint: you do not need to find a matrix representing the linear transformation.)
(a) $\mathcal P : \Bbb R^3 \rightarrow \Bbb R^3$ given by projection onto the plane $3x-y-2z=0$
(b) $\mathcal Q : M(2,2) \rightarrow M(2,2)$ given by $Q(A)=A^T$ (the transpose of A.)
(c) $\mathcal R : \mathcal P_2 \rightarrow \mathcal P_2$ given by $\mathcal R(p)=(x-2)p'(x)$
(where $\mathcal P_2 =$ the vector space of polynomials in $x$ of degree $\le 2.$)
This is a question I found in a practice exam which I was attempting. I was absolutely at odds with what to do, especially given the hint. Once I finished the exam, I came back to the question and found the matrix for the first question, which is
$$ \frac{1}{14}\begin{bmatrix} 5 & 3 & 6 \\ 3 & 13 & -2 \\ 6 & -2 & 10 \\ \end{bmatrix} $$
and the eigenvalues for that are $1,1,0$ with corresponding eigenvectors $(\frac{2}{3},0,1),(\frac{1}{3},1,0)$ and $(\frac{-3}{2},\frac{1}{2},1)$. I did this with the hope that I would be able to work backwards and see how they actually wanted me to find the answer so I could do it for the next parts. Given the hint and the very low mark value of this question on the paper, it can't be very complicated, but I'm stuck.
Please give me some hints as to how to proceed and tackle this question.
i think the question is asking for the exercise of "intuitive" perceptions. e.g. a projection clearly has eigen-values $0,1$ (for vectors perpendicular to or (resp) lying in the invariant subspace). symmetric and anti-symmetric matrices show that the transpose has eigenvalues $\pm 1$