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:
- $\R^n \to \R^m: \mathbf x \mapsto \mathbf {A x}$
- $\map {\mathrm N} {\mathbf A} = \set {\mathbf 0}$
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.
$\blacksquare$
Also see
Sources
- For a video presentation of the contents of this page, visit the Khan Academy.