The Rank of some [[Matrix]] is the number of rows with leading terms in its [[Reduced Row Echelon Form|RREF]] form. The more formal definition of the [[Rank]] of a [[Matrix]] $A$ is the [[Dimension]] of the [[Column Space]] of $A$. $\huge \op{rank}(A) = \dim \op{Col}\pa{A} $ [[Universal Quantifier|For any]] $A\in M_{n\times m}$, $\op{rank}(A) \le \op{min}(n, m)$ For any square [[Matrix]] $A\in M_{n\times n}$: | $\op{rank}(A) < n$ | $\op{rank}(A) = n$ | | ---------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------- | | $\op{span}(\set{\vec A_{1}, \dots, \vec A_{n}}) \neq \Rn n$ | $\op{span}(\set{\vec A_{1}, \dots, \vec A_{n}})$ **may** $\Rn n$ | | $\set{\vec A_{1}, \dots, \vec A_{n}}$ is [[Linear Dependence\|Linearly Dependent]] | <br>$\set{\vec A_{1}, \dots, \vec A_{n}}$ **may** [[Linear Independence\|Linearly Independent]] |