Editing Cokernel (section)

==Intuition==
The cokernel can be thought of as the space of ''constraints'' that an equation must satisfy, as the space of ''obstructions'', just as the [[Kernel (algebra)|kernel]] is the space of ''solutions.''

Formally, one may connect the kernel and the cokernel of a map {{math|''T'':  ''V'' →  ''W''}} by the [[exact sequence]]

:<math>0 \to \ker T \to V \overset T \longrightarrow W \to \operatorname{coker} T \to 0.</math>

These can be interpreted thus: given a linear equation {{math|1=''T''(''v'') = ''w''}} to solve,
* the kernel is the space of ''solutions'' to the ''homogeneous'' equation {{math|1=''T''(''v'') = 0}}, and its dimension is the number of ''degrees of freedom'' in solutions to {{math|1=''T''(''v'') = ''w''}}, if they exist;
* the cokernel is the space of ''constraints'' on ''w'' that must be satisfied if the equation is to have a solution, and its dimension is the number of independent constraints that must be satisfied for the equation to have a solution.

The dimension of the cokernel plus the dimension of the image (the rank) add up to the dimension of the target space, as the dimension of the quotient space {{math|''W'' / ''T''(''V'')}} is simply the dimension of the space ''minus'' the dimension of the image.

As a simple example, consider the map {{math|''T'': '''R'''<sup>2</sup> → '''R'''<sup>2</sup>}}, given by {{math|1=''T''(''x'', ''y'') = (0, ''y'')}}. Then for an equation {{math|1=''T''(''x'', ''y'') = (''a'', ''b'')}} to have a solution, we must have {{math|1=''a'' = 0}} (one constraint), and in that case the solution space is {{math|(''x'', ''b'')}}, or equivalently, {{math|1=(0, ''b'') + (''x'', 0)}}, (one degree of freedom). The kernel may be expressed as the subspace {{math|(''x'', 0) ⊆ ''V''}}: the value of {{mvar|x}} is the freedom in a solution. The cokernel may be expressed via the real valued map {{math|''W'': (''a'', ''b'') → (''a'')}}: given a vector {{math|(''a'', ''b'')}}, the value of {{mvar|a}} is the ''obstruction'' to there being a solution.

Additionally, the cokernel can be thought of as something that "detects" [[surjection]]s in the same way that the kernel "detects" [[injection (mathematics)|injection]]s. A map is injective if and only if its kernel is trivial, and a map is surjective if and only if its cokernel is trivial, or in other words, if {{math|1=''W'' = im(''T'')}}.