Linear Transformation Maps Zero Vector to Zero Vector

Theorem

Let $\mathbf V$ be a vector space, with zero $\mathbf 0$.

Likewise let $\mathbf V\,'$ be another vector space, with zero $\mathbf 0'$.

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

Then:

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

Corollary

$\mathbf 0 \in \map \ker T$

where $\map \ker T$ is the kernel of $T$.

Proof 1

From the vector space axioms we have that $\exists \mathbf 0 \in \mathbf V$.

It remains to be proved that $\map T {\mathbf 0} = \mathbf 0'$:

 $\ds \map T {\mathbf 0}$ $=$ $\ds \map T {\mathbf 0 + \mathbf 0}$ $\ds$ $=$ $\ds \map T {\mathbf 0} + \map T {\mathbf 0}$ Definition of Linear Transformation on Vector Space $\ds \leadsto \ \$ $\ds \mathbf 0'$ $=$ $\ds \map T {\mathbf 0}$ subtracting $\map T {\mathbf 0}$ from both sides

$\blacksquare$

Proof 2

From the vector space axioms we have that $\exists \mathbf 0 \in \mathbf V$.

What remains is to prove that $\map T {\mathbf 0} = \mathbf 0'$:

 $\ds \map T {\mathbf 0}$ $=$ $\ds \map T {0 \, \mathbf 0}$ Zero Vector Scaled is Zero Vector $\ds$ $=$ $\ds 0 \, \map T {\mathbf 0}$ Definition of Linear Transformation on Vector Space $\ds$ $=$ $\ds \mathbf 0'$ Vector Scaled by Zero is Zero Vector

$\blacksquare$