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Concept
Dimension (vector space)
Summary
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. 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 is if the dimension of is finite, and if its dimension is infinite.
The dimension of the vector space over the field can be written as or as read "dimension of over ". When can be inferred from context, is typically written.
The vector space has
as a standard basis, and therefore More generally, and even more generally, for any field
The complex numbers are both a real and complex vector space; we have and So the dimension depends on the base field.
The only vector space with dimension is the vector space consisting only of its zero element.
If is a linear subspace of then
To show that two finite-dimensional vector spaces are equal, the following criterion can be used: if is a finite-dimensional vector space and is a linear subspace of with then
The space has the standard basis where is the -th column of the corresponding identity matrix. Therefore, has dimension
Any two finite dimensional vector spaces over 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 is some set, a vector space with dimension over can be constructed as follows: take the set of all functions such that for all but finitely many in These functions can be added and multiplied with elements of to obtain the desired -vector space.
An important result about dimensions is given by the rank–nullity theorem for linear maps.
If is a field extension, then is in particular a vector space over Furthermore, every -vector space is also a -vector space.
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Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
Hilbert space
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