Rank

Let $T:V\to W$ be a linear map between vector spaces over a field $\mathbb{F}$. The rank of $T$ is

where $\operatorname{im}T=\{T(v):v\in V\}\subseteq W$.

If $A$ is an $m\times n$ matrix over $\mathbb{F}$, its rank is the rank of the associated linear map $x\mapsto Ax$, i.e.

Equivalent characterizations (matrices)

For an $m\times n$ matrix $A$, the following numbers are equal:

• $\dim($span of the columns of $A)$,
• $\dim($span of the rows of $A)$,
• the number of pivots in a row-reduced echelon form of $A$,
• the largest integer $r$ such that $A$ has an $r\times r$ minor with nonzero determinant.

In particular,

Rank–nullity

If $V$ is finite-dimensional, the kernel (null space) $\ker T = \{v\in V : T(v)=0\}$ satisfies

Example

For

the second row is $2$ times the first, so the row (and column) spaces are $1$-dimensional, hence $\operatorname{rank}(A)=1$.