In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and more generally to submanifolds in manifolds, and suitable subsets of algebraic varieties. Mathematics is the study of quantity, structure, space and change. ... In mathematics, if a set with certain properties is called a space, then a subset of which with same properties is usually called a subspace. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... This is a glossary of terms specific to differential geometry and differential topology. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... This article is about algebraic varieties. ...

If W is a vector space of dimension n, and V a linear subspace of W of dimension k, then the codimension of V is The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...

n − k.

The fundamental property of codimension lies in its relation to intersection: if W₁ has codimension k₁, and W₂ has codimension k₂, then if U is their intersection with codimension j we have In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

max (k₁, k₂) ≤ j ≤ k₁ + k₂.

In fact j may take any integer value in this range. This statement is more perspicuous than the translation in terms of dimensions, because the RHS is just the sum of the codimensions. In words The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, LHS is informal shorthand for the left-hand side of an equation. ...

codimensions (at most) add.

In terms of the dual space, it is quite evident why that is. The subspaces being defined by the vanishing of a certain number of linear functionals, which we can take to be linearly independent, that number is the codimension. Therefore we see that U is defined by taking the union of the sets of linear functionals defining the W_i. 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. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In linear algebra, a branch of mathematics, a linear functional is a linear function from a vector space to its field of scalars. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

In other language, which is basic for any kind of intersection theory, we are taking the union of a certain number of constraints. We have two phenomena to look out for: A constraint is a limitation of possibilities. ...

1. the two sets of constraints may not be independent;
2. the two sets of constraints may not be compatible.

The first of these is often expressed as the principle of counting constants: if we have a number N of parameters to adjust, 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. We do not expect to be able to find a solution if the predicted codimension, i.e. the number of independent constraints, exceeds N (in the linear algebra case, there is always a trivial, null vector solution, which is therefore discounted). A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ... A parameter is a measurement or value on which something else depends. ... In mathematics, a solution set for a collection of polynomials over some ring is defined to be the set . ... The term null vector can have two different meanings: null vector (vector space) null vector (Minkowski space) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

The second is a matter of geometry, on the model of parallel lines; it is something that can be discussed for linear problems by methods of linear algebra, and for non-linear problems in projective space, over the complex number field. Parallel Lines is a seminal New Wave album by the art punk band Blondie, released in September of 1978 (see 1978 in music). ... In mathematics, a projective space is a fundamental construction from any vector space. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...

Codimension also has some clear meaning in geometric topology: on a manifold (real) codimension 1 is the dimension of topological disconnection by a submanifold, while codimension 2 is the dimension of ramification and knot theory. In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... In mathematics, ramification is a geometric term used for branching out, in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. ... Trefoil knot, the simplest non-trivial knot. ...

See also: glossary of differential geometry and topology. This is a glossary of terms specific to differential geometry and differential topology. ...