Linear Transformation Maps Zero Vector to Zero Vector/Proof 2

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:

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

Proof

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

$\blacksquare$