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.
$\blacksquare$