projection from linear subspace algebraic geometry

the number of independent constraints, exceeds N (in the linear algebra case, there is always a trivial, null vector solution, which is therefore discounted). We do not expect to be able to find a solution if the predicted codimension, i.e. we have N degrees of freedom), and a constraint means we have to 'consume' a parameter to satisfy it, then the codimension of the solution set is at most the number of constraints. The first of these is often expressed as the principle of counting constraints: if we have a number N of parameters to adjust (i.e. the two sets of constraints may not be compatible.the two sets of constraints may not be independent.In other language, which is basic for any kind of intersection theory, we are taking the union of a certain number of constraints. This definition of codimension in terms of the number of functions needed to cut out a subspace extends to situations in which both the ambient space and subspace are infinite dimensional. That union may introduce some degree of linear dependence: the possible values of j express that dependence, with the RHS sum being the case where there is no dependence. Therefore, we see that U is defined by taking the union of the sets of linear functionals defining the W i. 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. In terms of the dual space, it is quite evident why dimensions add.

This statement is called dimension counting, particularly in intersection theory. If the subspaces or submanifolds intersect transversally (which occurs generically), codimensions add exactly. This statement is more perspicuous than the translation in terms of dimensions, because the RHS is just the sum of the codimensions. In fact j may take any integer value in this range. The fundamental property of codimension lies in its relation to intersection: if W 1 has codimension k 1, and W 2 has codimension k 2, then if U is their intersection with codimension j we have Īnd is dual to the relative dimension as the dimension of the kernel.įinite-codimensional subspaces of infinite-dimensional spaces are often useful in the study of topological vector spaces.Īdditivity of codimension and dimension counting If W is a linear subspace of a finite-dimensional vector space V, then the codimension of W in V is the difference between the dimensions: codim ⁡ ( W ) = dim ⁡ ( V ) − dim ⁡ ( W ). There is no “codimension of a vector space (in isolation)”, only the codimension of a vector subspace. For this reason, the height of an ideal is often called its codimension.Ĭodimension is a relative concept: it is only defined for one object inside another. In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.įor affine and projective algebraic varieties, the codimension equals the height of the defining ideal. Difference between the dimensions of mathematical object and a sub-object

Projection from linear subspace algebraic geometry