In mathematics, specifically

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...

and

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, relative dimension is the dual notion to codimension. In linear algebra, given a

quotient map In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...

V \to Q

, the difference dim ''V'' − dim ''Q'' is the relative dimension; this equals the dimension of the

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...

. 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 The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...

, while the relative dimension of a map is the dimension of the

. 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 In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often w ...

. Just as codimension is mostly used for injective maps, relative dimension is mostly used for surjective maps.

References

Algebraic geometry Geometric topology Linear algebra Dimension {{geometry-stub