Linear and multilinear algebra; matrix theory Definition

Multiplication by a matrix as a linear map

Builds on Matrix–vector product Linear map

Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$ . The linear map of multiplication by $A$ is $L_A \colon F^n \to F^m, \qquad L_A(x) = Ax,$ where $Ax$ is the matrix–vector product.

Remarks

The map $L_A$ is linear, because $A(x + y) = Ax + Ay$ and $A(cx) = c\, Ax$ . Its matrix in the standard bases is $A$ , because $L_A(e_j)$ is column $j$ of $A$ . Each linear map $F^n \to F^m$ equals $L_A$ for exactly one matrix $A$ . The kernel of $L_A$ is the null space of $A$ , and its image is the column space of $A$ .

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Matrix definition
Vector space definition
Matrix–vector product definition
Linear map definition
Multiplication by a matrix as a linear map definition

Tag	`0AJ0KP1B31`
URI	`m2://0AJ0KP1B31`
Kind	definition
Writing	W0 stub
MSC	15A04
Defines	Multiplication by a matrix as a linear map
Uses	Matrix–vector product, Linear map

Multiplication by a matrix as a linear map

Definition

Remarks

Reading path

Further reading

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