Editing Linear map (section)

===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>