Linear Transformation Maps Zero Vector to Zero Vector

Theorem
Let $\mathbf V$ be $\R^n$, or any other vector space, with a zero $\mathbf 0$.

Likewise let $\mathbf V\,'$ be $\R^m$, or any other vector space, with a zero $\mathbf 0\,'$.

Let $T: \mathbf V \to \mathbf V\,'$ be a linear transformation.

Then:


 * $T: \mathbf 0 \mapsto \mathbf 0\,'$

Corollary

 * $\mathbf 0 \in \ker \left({T}\right)$

where $\ker \left({ T }\right)$ is the kernel of $T$.

Proof
That $\exists \mathbf 0 \in \mathbf V$ follows from the vector space axioms.

What remains is to prove that $T\left({\mathbf 0}\right) = \mathbf 0\,'$:

Proof of Corollary
Follows from the main result and the definition of kernel.