Linear and multilinear algebra; matrix theory Definition

Kernel of a linear map

Definition

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. The kernel of $T$ is $\ker T = \{v \in V : T(v) = 0\}.$

Remarks

The kernel is a linear subspace of $V$ . A linear map is injective if and only if its kernel is trivial. For the map $x \mapsto Ax$ of a matrix $A$ , the kernel is the null space of $A$ , the solution set of $Ax = 0$ . The kernel of the projection $\R^3 \to \R^2$ , $(x, y, z) \mapsto (x, y)$ , is the $z$ -axis.

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Vector space definition
Linear map definition
Kernel of a linear map definition

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Tag	`02YFME3RGW`
URI	`m2://02YFME3RGW`
Kind	definition
Writing	W1 stated
MSC	15A04
Defines	Kernel of a linear map
Uses	Linear map

Kernel of a linear map

Definition

Remarks

Reading path

Where this leads

Further reading

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