Editing Linear map (section)

==Cokernel==
{{Main|Cokernel}}

A subtler invariant of a linear transformation <math display="inline">f: V \to W</math> is the [[cokernel|''co''kernel]], which is defined as
<math display="block">\operatorname{coker}(f) := W/f(V) = W/\operatorname{im}(f).</math>

This is the ''dual'' notion to the kernel: just as the kernel is a ''sub''space of the ''domain,'' the co-kernel is a [[quotient space (linear algebra)|''quotient'' space]] of the ''target.'' Formally, one has the [[exact sequence]]
<math display="block">0 \to \ker(f) \to V \to W \to \operatorname{coker}(f) \to 0.</math>

These can be interpreted thus: given a linear equation ''f''('''v''') = '''w''' to solve,

* the kernel is the space of ''solutions'' to the ''homogeneous'' equation ''f''('''v''') = 0, and its dimension is the number of [[degrees of freedom]] in the space of solutions, if it is not empty;
* the co-kernel is the space of [[wikt:constraint|constraints]] that the solutions must satisfy, and its dimension is the maximal number of independent constraints.

The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space ''W''/''f''(''V'') is the dimension of the target space minus the dimension of the image.

As a simple example, consider the map ''f'': '''R'''<sup>2</sup> → '''R'''<sup>2</sup>, given by ''f''(''x'', ''y'') = (0, ''y''). Then for an equation ''f''(''x'', ''y'') = (''a'', ''b'') to have a solution, we must have ''a'' = 0 (one constraint), and in that case the solution space is (''x'', ''b'') or equivalently stated, (0, ''b'') + (''x'', 0), (one degree of freedom). The kernel may be expressed as the subspace (''x'', 0) < ''V'': the value of ''x'' is the freedom in a solution – while the cokernel may be expressed via the map ''W'' → '''R''', <math display="inline"> (a, b) \mapsto (a)</math>: given a vector (''a'', ''b''), the value of ''a'' is the ''obstruction'' to there being a solution.

An example illustrating the infinite-dimensional case is afforded by the map ''f'': '''R'''<sup>∞</sup> → '''R'''<sup>∞</sup>, <math display="inline">\left\{a_n\right\} \mapsto \left\{b_n\right\}</math> with ''b''<sub>1</sub> = 0 and ''b''<sub>''n'' + 1</sub> = ''a<sub>n</sub>'' for ''n'' > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same [[cardinal number#Cardinal addition|sum]] as the rank and the dimension of the co-kernel (<math display="inline">\aleph_0 + 0 = \aleph_0 + 1</math>), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an [[endomorphism]] have the same dimension (0 ≠ 1). The reverse situation obtains for the map ''h'': '''R'''<sup>∞</sup> → '''R'''<sup>∞</sup>, <math display="inline">\left\{a_n\right\} \mapsto \left\{c_n\right\}</math> with ''c<sub>n</sub>'' = ''a''<sub>''n'' + 1</sub>. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.

===Index===
For a linear operator with finite-dimensional kernel and co-kernel, one may define ''index'' as:
<math display="block">\operatorname{ind}(f) := \dim(\ker(f)) - \dim(\operatorname{coker}(f)),</math>
namely the degrees of freedom minus the number of constraints.

For a transformation between finite-dimensional vector spaces, this is just the difference dim(''V'') − dim(''W''), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom.

The index of an operator is precisely the [[Euler characteristic]] of the 2-term complex 0 → ''V'' → ''W'' → 0. In [[operator theory]], the index of [[Fredholm operator]]s is an object of study, with a major result being the [[Atiyah–Singer index theorem]].<ref>{{SpringerEOM|title=Index theory|id=Index_theory&oldid=23864|first=Victor|last=Nistor}}: "The main question in index theory is to provide index formulas for classes of Fredholm operators ... Index theory has become a subject on its own only after M. F. Atiyah and I. Singer published their index theorems"</ref>