Rank

Number of non-zero rows in the row echelon form of $A$. Denoted $\text{Rank}\;A$.

Geometrically, $\text{Rank}\;A$ equals the dimension of the column space (image) of $A$: the number of linearly independent columns.

Properties

• $\text{Rank}\;A = \text{Rank}\;A^T$
• $\text{Rank }(AB) \le \min(\text{Rank}\;A,\; \text{Rank}\;B)$
• $\text{Rank}\;A \le \min(m, n)$