---
tag: 0EP73HK4C4
kind: definition
name: Row space
msc: [15A03]
defines: 069MCGX635
uses: [0HR573CHNM, 0N1MY1T0W9]
sources:
  - url: https://en.wikipedia.org/wiki/Row_and_column_spaces
    locator: Row space
---

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$ with rows $r_1, \dots, r_m \in F^n$.
The *row space* of $A$ is
$$\operatorname{Row}(A) = \operatorname{span}(r_1, \dots, r_m) \subseteq F^n,$$
the [[linear span|0DTY8B9HMS]] of the rows.

## Remarks

The row space of $A$ is the [[column space|03AJKXH1AV]] of $A^T$.
[[Elementary row operations|04XWH28452]] do not change the row space, so [[row equivalent|006BHFWMAP]] matrices have the same row space.
The nonzero rows of a row echelon form of $A$ form a basis of $\operatorname{Row}(A)$.
Its dimension equals the dimension of the column space, see [[row rank equals column rank|0TZAG60ZYT]].

## Explanations

### L1 human

The row space collects all the combinations of the rows of a matrix.
Row operations only mix the rows, so they keep the row space.
After row reduction, the nonzero rows give a basis of it.