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 $T(0_V) = 0_W$?
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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 $$0=T(v)-T(v)=(T(0)+T(v))-T(v)=T(0)+(T(v)-T(v))=T(0)+0=T(0).$$