finite-dimensional

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the dimension of a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
''V'' is the
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(i.e. the number of vectors) of a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
of ''V'' over its base
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
. p. 44, §2.36 It is sometimes called Hamel dimension (after
Georg Hamel Georg Karl Wilhelm Hamel (12 September 1877 – 4 October 1954) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens o ...

) or algebraic dimension to distinguish it from other types of
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

. 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
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
, and if its dimension is
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...

. The dimension of the vector space $V$ over the field $F$ can be written as $\dim_F\left(V\right)$ or as read "dimension of $V$ over $F$". When $F$ can be inferred from context, $\dim\left(V\right)$ is typically written.

# Examples

The vector space $\R^3$ has $\left\$ as a
standard basis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, and therefore $\dim_\left(\R^3\right) = 3.$ More generally, $\dim_\left(\R^n\right) = n,$ and even more generally, $\dim_\left(F^n\right) = n$ for any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
$F.$ The
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s $\Complex$ are both a real and complex vector space; we have $\dim_\left(\Complex\right) = 2$ and $\dim_\left(\Complex\right) = 1.$ So the dimension depends on the base field. The only vector space with dimension $0$ is $\,$ the vector space consisting only of its zero element.

# Properties

If $W$ is a
linear subspace In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of $V$ then $\dim \left(W\right) \leq \dim \left(V\right).$ 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 $\dim \left(W\right) = \dim \left(V\right),$ then $W = V.$ The space $\R^n$ has the standard basis $\left\,$ where $e_i$ is the $i$-th column of the corresponding
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

. Therefore, $\R^n$ has dimension $n.$ Any two finite dimensional vector spaces over $F$ with the same dimension are
isomorphic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. Any
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 $F^\left(B\right)$ of all functions $f : B \to F$ such that $f\left(b\right) = 0$ 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 The rank–nullity theorem is a theorem in 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 ...
for
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s. If $F / K$ is a
field extension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, 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 $\dim_K(V) = \dim_K(F) \dim_F(V).$ 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 $\dim V,$ then: :If dim $V$ is finite then $, V, = , F, ^.$ :If dim $V$ is infinite then $, V, = \max \left(, F, , \dim V\right).$

# Generalizations

A vector space can be seen as a particular case of a
matroid In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid Axiomatic system, axiomatically, the most si ...
, and in the latter there is a well-defined notion of dimension. The
length of a module In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
and the
rank of an abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
both have several properties similar to the dimension of vector spaces. The
Krull dimension In commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with ...
of a commutative ring, named after
Wolfgang Krull Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ...
(1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain of
prime ideal In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
s in the ring.

## Trace

The dimension of a vector space may alternatively be characterized as the
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * The Trace (album), ''The Trace'' (album) Other ...
of the
identity operator Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...
. For instance, $\operatorname\ \operatorname_ = \operatorname \left\left(\begin 1 & 0 \\ 0 & 1 \end\right\right) = 1 + 1 = 2.$ 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 Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
$A$ with maps $\eta : K \to A$ (the inclusion of scalars, called the ''unit'') and a map $\epsilon : A \to K$ (corresponding to trace, called the ''
counit In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
''). The composition $\epsilon \circ \eta : K \to K$ 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
bialgebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, this map is required to be the identity, which can be obtained by normalizing the counit by dividing by dimension ($\epsilon := \textstyle \operatorname$), 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 classIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
operators" on a
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, or more generally
nuclear operator In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a mathematician who became the leading figure in the cr ...
s on a
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. 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 Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, where the
character Character(s) may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to Theophrastus M ...
of a representation is the trace of the representation, hence a scalar-valued function on a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
$\chi : G \to K,$ whose value on the identity $1 \in G$ is the dimension of the representation, as a representation sends the identity in the group to the identity matrix: $\chi\left(1_G\right) = \operatorname\ I_V = \dim V.$ The other values $\chi\left(g\right)$ 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
: the $j$-invariant is the graded dimension of an infinite-dimensional graded representation of the
monster group In the area of abstract algebra known as group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő R ...
, and replacing the dimension with the character gives the McKay–Thompson series for each element of the Monster group.

* * * * * , also called Lebesgue covering dimension

# Sources

*

MIT Linear Algebra Lecture on Independence, Basis, and Dimension by Gilbert Strang
at MIT OpenCourseWare {{DEFAULTSORT:Dimension (Vector Space) Dimension Linear algebra Vectors (mathematics and physics)