Linear and multilinear algebra; matrix theory Proposition

A linear map is injective iff its kernel is trivial

Builds on Kernel of a linear map Linear map

Statement

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$ , and let $T \colon V \to W$ be a linear map. Then $T$ is injective if and only if $\ker T = \{0\}$ , where $\ker T$ is the kernel of $T$ .

Remarks

The key step is $T(u) = T(v) \iff T(u - v) = 0 \iff u - v \in \ker T$ . Thus each nonempty fibre $T^{-1}(w)$ is a coset $v + \ker T$ . For a matrix $A$ , the map $x \mapsto Ax$ is injective if and only if $Ax = 0$ has only the trivial solution.

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Vector space definition
Linear map definition
Kernel of a linear map definition
A linear map is injective iff its kernel is trivial statement

Tag	`0BJBYCGMWE`
URI	`m2://0BJBYCGMWE`
Kind	statement, proposition
Writing	W0 stub
MSC	15A04
Status	This copy of the site has no build data. Run `m2 status 0BJBYCGMWE`.
Uses	Kernel of a linear map, Linear map

A linear map is injective iff its kernel is trivial

Statement

Remarks

Reading path

Further reading

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