Linear Transformation is Injective iff Kernel Contains Only Zero/Corollary

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

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.

Also see

 * Null Space Contains Only Zero Vector iff Columns are Independent