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 of the rows.

Remarks

The row space of $A$ is the column space of $A^T$. Elementary row operations do not change the row space, so row equivalent 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.

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.