Linear Maps Take 0 to 0

Linear maps take $0$ to $0$

Suppose $T$ is a linear map from $V$ to $W$. Then $T(0)=0$.

Proof.

By additivity, we have

$$T(0)=T(0+0)=T(0)+T(0)$$

Add the additive inverse of $T(0)$ to each side of the equation above to conclude $T(0)=0$.