# Kernel of Linear Transformation contains Zero Vector

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## Corollary to Linear Transformation Maps Zero Vector to Zero Vector

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:

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

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

## Proof

Follows from Linear Transformation Maps Zero Vector to Zero Vector and the definition of kernel.

$\blacksquare$