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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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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