In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e. the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension.
For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say $V$ is if the dimension of $V$ is wiktionary:finite, finite, and if its dimension is infinity, infinite.
The dimension of the vector space $V$ over the field $F$ can be written as $\backslash dim\_F(V)$ or as $[V\; :\; F],$ read "dimension of $V$ over $F$". When $F$ can be inferred from context, $\backslash dim(V)$ is typically written.

MIT Linear Algebra Lecture on Independence, Basis, and Dimension by Gilbert Strang

at MIT OpenCourseWare {{DEFAULTSORT:Dimension (Vector Space) Dimension Linear algebra Vector spaces, Vectors (mathematics and physics)

Examples

The vector space $\backslash R^3$ has $$\backslash left\backslash $$ as a standard basis, and therefore $\backslash dim\_(\backslash R^3)\; =\; 3.$ More generally, $\backslash dim\_(\backslash R^n)\; =\; n,$ and even more generally, $\backslash dim\_(F^n)\; =\; n$ for any Field (mathematics), field $F.$ The complex numbers $\backslash Complex$ are both a real and complex vector space; we have $\backslash dim\_(\backslash Complex)\; =\; 2$ and $\backslash dim\_(\backslash Complex)\; =\; 1.$ So the dimension depends on the base field. The only vector space with dimension $0$ is $\backslash ,$ the vector space consisting only of its zero element.Properties

If $W$ is a linear subspace of $V$ then $\backslash dim\; (W)\; \backslash leq\; \backslash dim\; (V).$ To show that two finite-dimensional vector spaces are equal, the following criterion can be used: if $V$ is a finite-dimensional vector space and $W$ is a linear subspace of $V$ with $\backslash dim\; (W)\; =\; \backslash dim\; (V),$ then $W\; =\; V.$ The space $\backslash R^n$ has the standard basis $\backslash left\backslash ,$ where $e\_i$ is the $i$-th column of the corresponding identity matrix. Therefore, $\backslash R^n$ has dimension $n.$ Any two finite dimensional vector spaces over $F$ with 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 $f\; :\; B\; \backslash to\; F$ such that $f(b)\; =\; 0$ for all but finitely many $b$ in $B.$ These functions can be added and multiplied with elements of $F$ to obtain the desired $F$-vector space. An important result about dimensions is given by the rank–nullity theorem for linear maps. 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 $$\backslash dim\_K(V)\; =\; \backslash dim\_K(F)\; \backslash dim\_F(V).$$ In particular, every complex vector space of dimension $n$ is a real vector space of dimension $2n.$ Some 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 and if the dimension of $V$ is denoted by $\backslash dim\; V,$ then: :If dim $V$ is finite then $,\; V,\; =\; ,\; F,\; ^.$ :If dim $V$ is infinite then $,\; V,\; =\; \backslash max\; (,\; F,\; ,\; \backslash dim\; V).$Generalizations

A vector space can be seen 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. The Krull dimension of a commutative Ring (algebra), ring, named after Wolfgang Krull (1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.Trace

The dimension of a vector space may alternatively be characterized as the Trace (linear algebra), trace of the identity operator. For instance, $\backslash operatorname\backslash \; \backslash operatorname\_\; =\; \backslash operatorname\; \backslash left(\backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\backslash right)\; =\; 1\; +\; 1\; =\; 2.$ This appears to be a circular definition, but it allows useful generalizations. Firstly, it allows for a definition of a notion of dimension when one has a trace but no natural sense of basis. For example, one may have an Algebra over a field, algebra $A$ with maps $\backslash eta\; :\; K\; \backslash to\; A$ (the inclusion of scalars, called the ''unit'') and a map $\backslash epsilon\; :\; A\; \backslash to\; K$ (corresponding to trace, called the ''counit''). The composition $\backslash epsilon\; \backslash circ\; \backslash eta\; :\; K\; \backslash to\; K$ is a scalar (being a linear operator on a 1-dimensional space) corresponds to "trace of identity", and gives a notion of dimension for an abstract algebra. In practice, in bialgebras, this map is required to be the identity, which can be obtained by normalizing the counit by dividing by dimension ($\backslash epsilon\; :=\; \backslash textstyle\; \backslash operatorname$), so in these cases the normalizing constant corresponds to dimension. Alternatively, it may be possible to take the trace of operators on an infinite-dimensional space; in this case a (finite) trace is defined, even though no (finite) dimension exists, and gives a notion of "dimension of the operator". These fall under the rubric of "trace class operators" on a Hilbert space, or more generally nuclear operators on a Banach space. A subtler generalization is to consider the trace of a ''family'' of operators as a kind of "twisted" dimension. This occurs significantly in representation theory, where the Character (mathematics), character of a representation is the trace of the representation, hence a scalar-valued function on a Group (mathematics), group $\backslash chi\; :\; G\; \backslash to\; K,$ whose value on the identity $1\; \backslash in\; G$ is the dimension of the representation, as a representation sends the identity in the group to the identity matrix: $\backslash chi(1\_G)\; =\; \backslash operatorname\backslash \; I\_V\; =\; \backslash dim\; V.$ The other values $\backslash chi(g)$ of the character can be viewed as "twisted" dimensions, and find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of monstrous moonshine: the j-invariant, $j$-invariant is the graded dimension of an infinite-dimensional graded representation of the monster group, and replacing the dimension with the character gives the McKay–Thompson series for each element of the Monster group.See also

* * * * * , also called Lebesgue covering dimensionNotes

References

Sources

*External links

MIT Linear Algebra Lecture on Independence, Basis, and Dimension by Gilbert Strang

at MIT OpenCourseWare {{DEFAULTSORT:Dimension (Vector Space) Dimension Linear algebra Vector spaces, Vectors (mathematics and physics)