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
Canonical name	RanknullityTheorem
Date of creation	2013-03-22 16:35:40
Last modified on	2013-03-22 16:35:40
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	7
Author	yark (2760)
Entry type	Theorem
Classification	msc 15A03
Related topic	RankLinearMapping
Related topic	Nullity