Editing Rank–nullity theorem (section)

== Reformulations and generalizations ==
This theorem is a statement of the [[first isomorphism theorem]] of algebra for the case of vector spaces; it generalizes to the [[splitting lemma]].

In more modern language, the theorem can also be phrased as saying that each short exact sequence of vector spaces splits. Explicitly, given that
<math display="block"> 0 \rightarrow U \rightarrow V \mathbin{\overset{T}{\rightarrow}} R \rightarrow 0 </math>
is a [[short exact sequence]] of vector spaces, then <math> U \oplus R \cong V </math>, hence
<math display="block">\dim(U) + \dim(R) = \dim(V) .</math>
Here <math>R</math> plays the role of <math>\operatorname{Im} T</math> and <math>U</math> is <math>\operatorname{Ker}T</math>, i.e.
<math display="block"> 0 \rightarrow \ker T \mathbin{\hookrightarrow} V \mathbin{\overset{T}{\rightarrow}} \operatorname{im} T \rightarrow 0</math>

In the finite-dimensional case, this formulation is susceptible to a generalization: if
<math display="block">0 \rightarrow V_1 \rightarrow V_2 \rightarrow \cdots V_r \rightarrow 0</math>
is an [[exact sequence]] of finite-dimensional vector spaces, then<ref>{{cite web|last1=Zaman|first1=Ragib|title=Dimensions of vector spaces in an exact sequence|url=https://math.stackexchange.com/q/255384|accessdate=27 October 2015|website=Mathematics Stack Exchange|ref=DimVS}}</ref>
<math display="block">\sum_{i=1}^r (-1)^i\dim(V_i) = 0.</math>
The rank–nullity theorem for finite-dimensional vector spaces may also be formulated in terms of the ''index'' of a linear map. The index of a linear map <math>T \in \operatorname{Hom}(V,W)</math>, where <math>V</math> and <math>W</math> are finite-dimensional, is defined by
<math display="block"> \operatorname{index} T = \dim \operatorname{Ker}(T) - \dim \operatorname{Coker} T .</math>

Intuitively, <math>\dim \operatorname{Ker} T</math> is the number of independent solutions <math>v</math> of the equation <math>Tv = 0</math>, and <math>\dim \operatorname{Coker} T </math> is the number of independent restrictions that have to be put on <math>w</math> to make <math>Tv = w </math> solvable. The rank–nullity theorem for finite-dimensional vector spaces is equivalent to the statement
<math display="block"> \operatorname{index} T = \dim V - \dim W . </math>

We see that we can easily read off the index of the linear map <math>T</math> from the involved spaces, without any need to analyze <math>T</math> in detail. This effect also occurs in a much deeper result: the [[Atiyah–Singer index theorem]] states that the index of certain differential operators can be read off the geometry of the involved spaces.