= Solution
For an $m\times n$ <matrix> $A$ with real entries, its <row space> is the <linear span> of its rows, a <linear subspace> of $\mathbb R^n$. Its <column space> is the <linear span> of its columns, a <linear subspace> of $\mathbb R^m$ and the <image of a linear map> $x\mapsto Ax$. Its <null space> is
$$
\ker A=\{x\in\mathbb R^n:Ax=0\},
$$
a <linear subspace> of $\mathbb R^n$. Finally,
$$
\boxed{\operatorname{rank}A=\dim\operatorname{Col}A=\dim\operatorname{Row}A.}
$$
Thus the <rank of a matrix> counts the number of <linearly independent> columns, equivalently rows. The <nullity> is $\dim\ker A=n-\operatorname{rank}A$ by the <rank-nullity theorem>.
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