Editing Codimension (section)

==Definition==
Codimension is a ''relative'' concept: it is only defined for one object ''inside'' another. There is no “codimension of a vector space (in isolation)”, only the codimension of a vector ''sub''space.

If ''W'' is a [[linear subspace]] of a [[finite-dimensional]] [[vector space]] ''V'', then the '''codimension''' of ''W'' in ''V'' is the difference between the dimensions:<ref>{{harvnb|Roman|2008|loc=p. 93 §3}}</ref>
:<math>\operatorname{codim}(W) = \dim(V) - \dim(W).</math>
It is the complement of the dimension of ''W,'' in that, with the dimension of ''W,'' it adds up to the dimension of the [[ambient space]] ''V:''
:<math>\dim(W) + \operatorname{codim}(W) = \dim(V).</math>

Similarly, if ''N'' is a submanifold or subvariety in ''M'', then the codimension of ''N'' in ''M'' is
:<math>\operatorname{codim}(N) = \dim(M) - \dim(N).</math>
Just as the dimension of a submanifold is the dimension of the [[tangent bundle]] (the number of dimensions that you can move ''on'' the submanifold), the codimension is the dimension of the [[normal bundle]] (the number of dimensions you can move ''off'' the submanifold).

More generally, if ''W'' is a [[linear subspace]] of a (possibly infinite dimensional) [[vector space]] ''V'' then the codimension of ''W'' in ''V'' is the dimension (possibly infinite) of the [[quotient space (linear algebra)|quotient space]] ''V''/''W'', which is more abstractly known as the [[cokernel]] of the inclusion. For finite-dimensional vector spaces, this agrees with the previous definition
:<math>\operatorname{codim}(W) = \dim(V/W) = \dim \operatorname{coker} ( W \to V ) = \dim(V) - \dim(W),</math>
and is dual to the relative dimension as the dimension of the [[kernel (algebra)|kernel]].

Finite-codimensional subspaces of infinite-dimensional spaces are often useful in the study of [[topological vector space]]s.