Matrix Rank

The column(/row) rank of a matrix \(\mathbf{A} \in \mathbb{R}^{n\times m}\) is the dimension of the subspace spanned by its columns(/rows).

It happens to be the case that the column rank and row rank are equal, hence it is often just called the rank.

The matrix \(\mathbf{A} \in \mathbb{R}^{n\times m}\) is said to have full rank if its rank is \(\min(n,m)\).