# rank-nullity theorem

Let $V$ and $W$ be vector spaces over the same field. If $\phi\colon V\to W$ is a linear mapping, then

 $\operatorname{dim}V=\operatorname{dim}(\operatorname{ker}\phi)+\operatorname{% dim}(\operatorname{im}\phi).$

In other words, the dimension of $V$ is equal to the sum (http://planetmath.org/CardinalArithmetic) of the rank (http://planetmath.org/RankLinearMapping) and nullity of $\phi$.

Note that if $U$ is a subspace of $V$, then this (applied to the canonical mapping $V\to V/U$) says that

 $\operatorname{dim}V=\operatorname{dim}U+\operatorname{dim}(V/U),$

that is,

 $\operatorname{dim}V=\operatorname{dim}U+\operatorname{codim}U,$

where $\operatorname{codim}$ denotes codimension.

An alternative way of stating the rank-nullity theorem is by saying that if

 $0\to U\to V\to W\to 0$

is a short exact sequence of vector spaces, then

 $\operatorname{dim}(V)=\operatorname{dim}(U)+\operatorname{dim}(W).$

In fact, if

 $0\to V_{1}\to\cdots\to V_{n}\to 0$

is an exact sequence of vector spaces, then

 $\sum_{i=1}^{\lfloor n/2\rfloor}V_{2i}=\sum_{i=1}^{\lceil n/2\rceil}V_{2i-1},$

that is, the sum of the dimensions of even-numbered terms is the same as the sum of the dimensions of the odd-numbered terms.

Title rank-nullity theorem RanknullityTheorem 2013-03-22 16:35:40 2013-03-22 16:35:40 yark (2760) yark (2760) 7 yark (2760) Theorem msc 15A03 RankLinearMapping Nullity