I am new to linear algebra, and cannot work out the following question, despite the fact that I have been thinking about it for a long while.
Let $V$ be a linear space of n dimensions over R, and let $S,T:V \to V$ be linear transformations.
True or False?
- If $v$ is an eigenvector of $S$ and of $T$, then $v$ is also an eigenvector of $S + T$.
- If $λ_1$ is an eigenvalue of $S$ and $λ_2$ is an eigenvalue of $T$, then $λ_1$ + $λ_2$ is a eigenvalue of $S + T$.
I am not just looking for the right answer, but also for the reasoning behind it…