Relative dimension

In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension.

In linear algebra, given a quotient map $$V \to Q$$, the difference dim V − dim Q is the relative dimension; this equals the dimension of the kernel.

In fiber bundles, the relative dimension of the map is the dimension of the fiber.

More abstractly, the codimension of a map is the dimension of the cokernel, while the relative dimension of a map is the dimension of the kernel.

These are dual in that the inclusion of a subspace $$V \to W$$ of codimension k dualizes to yield a quotient map $$W^* \to V^*$$ of relative dimension k, and conversely.

The additivity of codimension under intersection corresponds to the additivity of relative dimension in a fiber product. Just as codimension is mostly used for injective maps, relative dimension is mostly used for surjective maps.