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In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension. All bases of a vector space have equal cardinality (see dimension theorem for vector spaces) and so the dimension of a vector space is uniquely defined. The dimension of the vector space V over the field F is written as dimF(V). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...
In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...
Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...
In mathematics, the dimension theorem for vector spaces states that a vector space has a definite, well-defined number of dimensions. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
We say V is finite-dimensional if the dimension of V is finite. In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
Examples
E.g. The vector space R3 has {(1,0,0), (0,1,0), (0,0,1)} as a basis, and therefore we have dimR(R3) = 3. More generally, dimR(Rn) = n. And more generally still, dimF(Fn) = n.
The complex numbers C are a real vector space; we have dimR(C) = 2 and dimC(C) = 1. So the dimension depends on the base field. In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
The only vector space with dimension 0 is {0}, the vector space consisting only of its zero element.
Facts If W is a linear subspace of V, then dim(W) ≤ dim(V). The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...
To show that two finite-dimensional vector spaces are equal, one often uses the following criterion: if V is a finite-dimensional vector space and W is a linear subspace of V with dim(W) = dim(V), then W = V.
Any two vector spaces over F having the same dimension are isomorphic. Any bijective map between their bases can be uniquely extended to a bijective linear map between the vector spaces. If B is some set, a vector space with dimension |B| over F can be constructed as follows: take the set F(B) of all functions f : B → F such that f(b) = 0 for all but finitely many b in B. These functions can be added and multiplied with elements of F, and we obtain the desired F-vector space. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...
An important result about dimensions related to a linear transformation is given by the rank-nullity theorem. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
In mathematics, the rank-nullity theorem of linear algebra, in its simplest form, relates the rank and the nullity of a matrix together with the number of columns of the matrix. ...
If F/K is a field extension, then F is in particular a vector space over K. Furthermore, every F-vector space V is also a K-vector space. The dimensions are related by the formula In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ...
- dimK(V) = dimK(F) dimF(V).
In particular, every complex vector space of dimension n is a real vector space of dimension 2n.
Some simple formulae relate the dimension of a vector space with the cardinality of the base field and the cardinality of the space itself. If V is a vector space over a field F then, denoting the dimension of V by dimV, we have:
- If dimV is finite, then |V| = |F|dimV.
- If dimV is infinite, then |V| = max(|F|, dimV).
Generalizations One can see a vector space as a particular case of a matroid, and in the latter there is a well defined notion of dimension. The length of a module and the rank of an abelian group both have several properties similar to the dimension of vector spaces. In combinatorial mathematics, a matroid is a structure that captures the essence of a notion of independence that generalizes linear independence in vector spaces. ...
In abstract algebra, the length of a module is a measure of the modules size. It is defined as the length of the longest ascending chain of submodules and is a generalization of the concept of dimension for vector spaces. ...
In mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space one would need to contain it; or alternatively how large a free abelian group it can contain as a subgroup. ...
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