Linear maps take 0 to 0 Suppose $T$ is a linear map from $V$ to $W$ where $V$ and $W$ are subspaces of a finite-dimensional vector space over some generic field $\mathcal{F}$.
Show $T(0)=0$.
Proof:
By additivity of a linear map,
(1) $T(0)=T(0+0)=T(0)+T(0)$.
Since $T(0)\in W$ and $W$ is a subspace, there exists an additive inverse for $T(0)$: denote as $k$.
Add $k$ on both sides of (1), we obtain
$T(0)=0$.
Is this proof correct?
Reference:
Axler, Sheldon J. $\textit{Linear Algebra Done Right}$, New York: Springer, 2015.

Well, given two groups $(G,\cdot,e)$ and $(G',\circ,e')$ (think of the additive groups of vector spaces) and a homomorphism $\phi:G\rightarrow G'$, i.e., $\phi$ is a mapping with
$\phi(g\cdot h) =\phi(g)\circ \phi(h)$.
Then $e'\circ \phi(e) = \phi(e) = \phi(e\cdot e) = \phi(e)\circ \phi(e)$. By multiplying with $\phi(e)^{-1}$ from the right, we obtain $e'=\phi(e)$. Done.
