Whether linear transformation maps $0$ to $0$

Question

Suppose I have a linear transformation $T: V\to W$. If I perform this transformation on the $0$ vector of $V$, $0_V$, does that necessarily mean its image will be $0_W$? In other words, is it necessarily true that $T(0_V) = 0_W$?

Answer 1

Yes. This is true. Note that $T(0\cdot 0_V)=0T(0_V)=0_W$.

Answer 2

We use only the usual axioms of a vector space and linear map to obtain $$T(v)=T(v+0)=T(0+v)=T(0)+T(v).$$ It follows that $$0=T(v)-T(v)=(T(0)+T(v))-T(v)=T(0)+(T(v)-T(v))=T(0)+0=T(0).$$