---
kind: statement
tag: ZE36VEAY80
name: Linearity of the matrix-vector product
role: proposition
msc:
  - 15A04
uses:
  - 0PDVQEP0ZC
about:
  - 0PDVQEP0ZC
sources:
  - url: https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section2-2.xml#L829
    locator: src/section2-2.xml:829
---

## 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|0PDVQEP0ZC]] 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 $c = 0$.