Proving either $T(v)=0$ or $T(x)=v$ has a solution for every $v$ in $V$?

Question

If $T:V\longrightarrow V$ is a linear transformation where $V$ is finite dimensional, then either:

a) $T(v)=0$, for some $v≠0$ in $V$.

b) $T(x)=v$ has a solution $x\in V$ for every $v\in V$.

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I don't think I properly understand the question. Please help me

Answer 1

If a) occurs, then $T$ is not injecive. Therefore, it is not surjective. That is, there is some $v\in V$ which is not in the range of $T$.

And if a) doesn't occur, then $T$ is injective and therefore surjective. So, b) occurs.