Rank of a matrix

Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$. The rank of $A$ is $$\rank A = \dim \operatorname{Col}(A),$$ the dimension of the column space of $A$.

Remarks

The rank is the maximal number of linearly independent columns of $A$. It also equals the dimension of the row space, by row rank equals column rank. It equals the number of pivots in a row echelon form of $A$, and the largest order of a nonzero minor of $A$. An $n \times n$ matrix is invertible if and only if its rank is $n$.

Explanations

L1 human

The rank of a matrix counts how many of its columns are really independent. For $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$, the second column is twice the first, so the rank is $1$. We compute the rank by row reduction and count the pivots.