Rank, Nullity, and the Four Subspaces

The column space $C(A)$ is where $A$ can send vectors: all outputs $A\mathbf{x}$. Its dimension is the rank. The null space $N(A)$ is what $A$ destroys: all $\mathbf{x}$ with $A\mathbf{x} = \mathbf{0}$. Its dimension is the nullity.

If 1000 data points in 50 dimensions actually lie in a 3-dimensional subspace, the data matrix has rank 3: only 3 independent directions of variation exist. Rank measures the "information content" of a matrix.