= Solution
For a real $m\times n$ <matrix> $A$, the <row space> is the <linear span> of its rows in $\mathbb R^n$. The <column space> is the <linear span> of its columns in $\mathbb R^m$, equivalently $\{A\mathbf x:\mathbf x\in\mathbb R^n\}$. The <null space> is
$$
\ker A=\{\mathbf x\in\mathbb R^n:A\mathbf x=\mathbf0\}.
$$
The <matrix rank> is the <dimension> of the <column space>, which equals the <dimension> of the <row space>. In particular,
$$
\boxed{\operatorname{rank}A=\dim\operatorname{Row}(A)=\dim\operatorname{Col}(A),\qquad
\dim\ker A=n-\operatorname{rank}A.}
$$
The last equality is the <rank-nullity theorem>.
Back to article page
