High Quality Content by WIKIPEDIA articles! In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and more generally to submanifolds in manifolds, and suitable subsets of algebraic varieties. 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.In terms of the dual space, it is quite evident why dimensions add. The subspaces can be defined by the vanishing of a certain number of linear functionals, which if we take to be linearly independent, their number is the codimension. Therefore we see that U is defined by taking the union of the sets of linear functionals defining the Wi.

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