row space

$\operatorname{Row}(A)$ — the span of a matrix's rows.

For an $m\times n$ matrix $A$ with rows $\mathbf r_1,\dots,\mathbf r_m$ , the row space $\operatorname{Row}(A) = \operatorname{span}\{\mathbf r_1,\dots,\mathbf r_m\}$ is a subspace of $\mathbb R^n$ (one coordinate per column), while $\operatorname{Col}(A)$ sits in $\mathbb R^m$ — the two live in different ambient spaces whenever $A$ isn't square. $\operatorname{Row}(A)=\operatorname{Col}(A^{\mathsf T})$ , and elementary row operations never change a matrix's row space.