Dimension (vector space)

Last updated

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field. [1] It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension.

Contents

For every vector space there exists a basis, [lower-alpha 1] and all bases of a vector space have equal cardinality; [lower-alpha 2] as a result, the dimension of a vector space is uniquely defined. We say V is finite-dimensional if the dimension of V is finite, and infinite-dimensional if its dimension is infinite.

The dimension of the vector space V over the field F can be written as dimF(V) or as [V : F], read "dimension of V over F". When F can be inferred from context, dim(V) is typically written.

Examples

The vector space R3 has

${\displaystyle \left\{{\begin{pmatrix}1\\0\\0\end{pmatrix}},{\begin{pmatrix}0\\1\\0\end{pmatrix}},{\begin{pmatrix}0\\0\\1\end{pmatrix}}\right\}}$

as a standard basis, and therefore we have dimR(R3) = 3. More generally, dimR(Rn) = n, and even more generally, dimF(Fn) = n for any field F.

The complex numbers C are both a real and complex vector space; we have dimR(C) = 2 and dimC(C) = 1. So the dimension depends on the base field.

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).

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.

Rn has the standard basis {e1, ..., en}, where ei is the i-th column of the corresponding identity matrix. Therefore, Rn has dimension n.

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 : BF 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.

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

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 dim V, we have:

If dim V is finite, then |V| = |F|dim V.
If dim V is infinite, then |V| = max(|F|, dim V).

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.

The Krull dimension of a commutative ring, named after Wolfgang Krull (18991971), 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 of the identity operator. For instance, ${\displaystyle \operatorname {tr} \ \operatorname {id} _{\mathbf {R} ^{2}}=\operatorname {tr} \left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)=1+1=2.}$ This appears to be a circular definition, but it allows useful generalizations.

Firstly, it allows one to define a notion of dimension when one has a trace but no natural sense of basis. For example, one may have an algebra A with maps ${\displaystyle \eta \colon K\to A}$ (the inclusion of scalars, called the unit) and a map ${\displaystyle \epsilon \colon A\to K}$ (corresponding to trace, called the counit ). The composition ${\displaystyle \epsilon \circ \eta \colon K\to K}$ 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 bialgebras one requires that this map be the identity, which can be obtained by normalizing the counit by dividing by dimension (${\displaystyle \epsilon$ :=\textstyle {\frac {1}{n}}\operatorname {tr} }), so in these cases the normalizing constant corresponds to dimension.

Alternatively, one may be able 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 of a representation is the trace of the representation, hence a scalar-valued function on a group ${\displaystyle \chi \colon G\to K,}$ whose value on the identity ${\displaystyle 1\in G}$ is the dimension of the representation, as a representation sends the identity in the group to the identity matrix: ${\displaystyle \chi (1_{G})=\operatorname {tr} \ I_{V}=\dim V.}$ One can view the other values ${\displaystyle \chi (g)}$ of the character 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 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. [2]

Notes

1. if one assumes the axiom of choice

Related Research Articles

In mathematics, a linear map is a mapping VW between two modules that preserves the operations of addition and scalar multiplication. If a linear map is a bijection then it is called a linear isomorphism.

In linear algebra, the trace of a square matrix A, denoted , is defined to be the sum of elements on the main diagonal of A.

In mathematics, a trace-class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and reserve "nuclear operator" for usage in more general topological vector spaces.

In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of vector spaces of finite dimension is the characteristic polynomial of the matrix of the endomorphism over any base; it does not depend on the choice of a basis. The characteristic equation, also known as the determinantal equation, is the equation obtained by equating to zero the characteristic polynomial.

In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras.

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.

The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank and its nullity.

In mathematics, the Heisenberg group, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form

In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a bounded operator, and so continuous.

In mathematics, an invariant subspace of a linear mapping T : VV from some vector space V to itself, is a subspace W of V that is preserved by T; that is, T(W) ⊆ W.

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.

In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in quantum information and decoherence which is relevant for quantum measurement and thereby to the decoherent approaches to interpretations of quantum mechanics, including consistent histories and the relative state interpretation.

The representation theory of groups is a part of mathematics which examines how groups act on given structures.

In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by Hermann Weyl. There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula.

In homological algebra, Whitehead's lemmas represent a series of statements regarding representation theory of finite-dimensional, semisimple Lie algebras in characteristic zero. Historically, they are regarded as leading to the discovery of Lie algebra cohomology.

In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of simple objects, and simple objects are those that do not contain non-trivial proper sub-objects. The precise definitions of these words depends on the context.

This is a glossary of representation theory in mathematics.

In mathematics, specifically in representation theory, a semisimple representation is a linear representation of a group or an algebra that is a direct sum of simple representations. It is an example of the general mathematical notion of semisimplicity.

In mathematics, the representation theory of semisimple Lie algebras is one of crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra ; in particular, it gives a way to parametrize irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight.

This is a glossary for the terminology in a mathematical field of functional analysis.

References

1. Itzkov, Mikhail (2009). Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics. Springer. p. 4. ISBN   978-3-540-93906-1.
2. Gannon, Terry (2006), Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, ISBN   0-521-83531-3