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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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Yes: $T(0)=T(v-v)=T(v)-T(v)=0$. – Clayton Dec 9 '12 at 17:38
up vote 3 down vote accepted

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

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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 $$0=T(v)-T(v)=(T(0)+T(v))-T(v)=T(0)+(T(v)-T(v))=T(0)+0=T(0).$$

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Congratulations! 10K rep!!! – amWhy Dec 13 '12 at 0:01

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