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Concept
Codimension
Summary
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension.
The dual concept is relative dimension.
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 subspace.
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:
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:
Similarly, if N is a submanifold or subvariety in M, then the codimension of N in M is
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 V/W, which is more abstractly known as the cokernel of the inclusion. For finite-dimensional vector spaces, this agrees with the previous definition
and is dual to the relative dimension as the dimension of the kernel.
Finite-codimensional subspaces of infinite-dimensional spaces are often useful in the study of topological vector spaces.
The fundamental property of codimension lies in its relation to intersection: if W1 has codimension k1, and W2 has codimension k2, then if U is their intersection with codimension j we have
max (k1, k2) ≤ j ≤ k1 + k2.
In fact j may take any integer value in this range.
Official source
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