Linear Transformation is Injective iff Kernel Contains Only Zero/Corollary

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Corollary to Linear Transformation is Injective iff Kernel Contains Only Zero

Let $\mathbf A$ be in the matrix space $\map {\mathbf M_{m, n} } \R$

Then the mapping:

$\R^n \to \R^m: \mathbf x \mapsto \mathbf {A x}$

is injective if and only if:

$\map {\mathrm N} {\mathbf A} = \set {\mathbf 0}$

where $\map {\mathrm N} {\mathbf A}$ is the null space of $\mathbf A$.


Proof

From Matrix Product as Linear Transformation, $\mathbf x \mapsto \mathbf {A x}$ defines a linear transformation.

The result follows from Linear Transformation is Injective iff Kernel Contains Only Zero and the definition of null space.

$\blacksquare$


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