I was reading a proof of the theorem that the range of a linear map $T$ is always a subspace of the target space, and when the author was showing that the $0$ vector was included in the range, he made an appeal to a previous theorem which says that the null space of $T$ is always a subspace of $T$.
In other words, he says that because the nullspace is a subspace, $0$ is always in the nullspace, and therefore since $T(0) = 0$, then $0$ is in the range of $T$.
That makes sense, but is it possible that the nullspace is empty? My feeling is no. Because $T$ acts on a vector space $V$, then $V$ must include $0$, and since we showed that the nullspace is a subspace, then $0$ is always in the nullspace of a linear map, so therefore the nullspace of a linear map can never be empty as it must always include at least one element, namely $0$.
Best Answer
Let $T:V\to W$ be a linear map. Then $$ T(\mathbf{0})=T(\mathbf{0}-\mathbf{0})=T(\mathbf{0})-T(\mathbf{0})=\mathbf{0} $$ This proves that $\mathbf 0$ is always in the nullspace of $T$. Hence the nullspace of $T$ cannot be empty.