Linear and multilinear algebra; matrix theory Proposition

Linearity of the matrix-vector product

Statement

Let $A$ be a matrix. Let $v$ and $w$ be column vectors whose size matches the number of columns of $A$ , so that the matrix-vector products below are defined. Let $c$ be a scalar. Then three facts hold.

First, $A0 = 0$ , where the zero vector on the left has the size of the columns of $A$ and the zero vector on the right has the size of the rows of $A$ .

Second, $A(cv) = c(Av)$ .

Third, $A(v + w) = Av + Aw$ .

Remarks

The second and third facts say that $v \mapsto Av$ is a linear map. The first fact follows from the second with $c = 0$ .

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Matrix–vector product concept
Linearity of the matrix-vector product statement

Tag	`ZE36VEAY80`
URI	`m2://ZE36VEAY80`
Kind	statement, proposition
Writing	W0 stub
MSC	15A04
Status	This copy of the site has no build data. Run `m2 status ZE36VEAY80`.
About	Matrix–vector product
Uses	Matrix–vector product
Changed	4 November 2026

Linearity of the matrix-vector product

Statement

Remarks

Reading path

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