Problem: Let $V$ be a vector space and $\mathcal{L} (V)$ be the set of all linear maps from $V$ to $V$. Let $T \in \mathcal{L} (V)$. Suppose that there exists nonzero vectors $\mathbf{v} , \mathbf{w} \in V$ such that
\[
T(\mathbf{v} ) = 5 \mathbf{w} \text{ and } T(\mathbf{w}) = 5 \mathbf{v}.
\]
Then show that $5$ or $-5$ is an eigenvalues of $T$.
Solution: I encourage you to attempt to solve the problem today. The solution will be provided tomorrow. This will give you the opportunity to test your understanding of the problem and to improve your skills in solving similar problems in the future.