= Solution
For a real $m\times n$ <matrix> $A$, the <row space> is the <linear span> of its rows in $\mathbb R^n$, and the <column space> is the <linear span> of its columns in $\mathbb R^m$. Its <null space> consists of the input <vectors> sent to zero:
$$
\operatorname{Null}(A)=\{x\in\mathbb R^n:Ax=0\}.
$$
The <rank of a matrix> is
$$
\boxed{\operatorname{rank}A=\dim\operatorname{Col}(A)=\dim\operatorname{Row}(A).}
$$
The equality of row and column dimensions is the row-rank/column-rank theorem. Each displayed space is a <linear subspace> of the indicated ambient <vector space>.
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