Relative dimension

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 o 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 o W of codimension "k" dualizes to yield a quotient map W^* o 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.


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