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[parent] rank-nullity theorem (Theorem)

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

$\displaystyle \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 of the rank 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

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

$\displaystyle 0\to U\to V\to W\to 0 $
is a short exact sequence of vector spaces, then
$\displaystyle \operatorname{dim}(V) = \operatorname{dim}(U) + \operatorname{dim}(W). $
In fact, if
$\displaystyle 0\to V_1\to\cdots\to V_n\to 0 $
is an exact sequence of vector spaces, then
$\displaystyle \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.



"rank-nullity theorem" is owned by yark.
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See Also: rank of a linear mapping, nullity


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another proof of rank-nullity theorem (Proof) by CWoo
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Cross-references: exact sequence, short exact sequence, codimension, mapping, subspace, nullity, dimension, linear mapping, field, vector spaces
There are 5 references to this entry.

This is version 4 of rank-nullity theorem, born on 2007-01-18, modified 2007-01-20.
Object id is 8790, canonical name is RankNullityTheorem2.
Accessed 1142 times total.

Classification:
AMS MSC: 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)
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