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][2] 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,^{[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 ${\displaystyle V}$ is finite-dimensional if the dimension of ${\displaystyle V}$ is finite, and infinite-dimensional if its dimension is infinite.

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

Examples

The vector space ${\displaystyle \mathbb {R} ^{3}}$ 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 ${\displaystyle \dim _{\mathbb {R} }(\mathbb {R} ^{3})=3.}$ More generally, ${\displaystyle \dim _{\mathbb {R} }(\mathbb {R} ^{n})=n,}$ and even more generally, ${\displaystyle \dim _{F}(F^{n})=n}$ for any field ${\displaystyle F.}$

The complex numbers ${\displaystyle \mathbb {C} }$ are both a real and complex vector space; we have ${\displaystyle \dim _{\mathbb {R} }(\mathbb {C} )=2}$ and ${\displaystyle \dim _{\mathbb {C} }(\mathbb {C} )=1.}$ So the dimension depends on the base field.

The only vector space with dimension ${\displaystyle 0}$ is ${\displaystyle \{0\},}$ the vector space consisting only of its zero element.

Properties

If ${\displaystyle W}$ is a linear subspace of ${\displaystyle V}$ then ${\displaystyle \dim(W)\leq \dim(V).}$

To show that two finite-dimensional vector spaces are equal, the following criterion can be used: if ${\displaystyle V}$ is a finite-dimensional vector space and ${\displaystyle W}$ is a linear subspace of ${\displaystyle V}$ with ${\displaystyle \dim(W)=\dim(V),}$ then ${\displaystyle W=V.}$

The space ${\displaystyle \mathbb {R} ^{n}}$ has the standard basis ${\displaystyle \left\{e_{1},\ldots ,e_{n}\right\},}$ where ${\displaystyle e_{i}}$ is the ${\displaystyle i}$-th column of the corresponding identity matrix. Therefore, ${\displaystyle \mathbb {R} ^{n}}$ has dimension ${\displaystyle n.}$

Any two finite dimensional vector spaces over ${\displaystyle F}$ 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 ${\displaystyle B}$ is some set, a vector space with dimension ${\displaystyle |B|}$ over ${\displaystyle F}$ can be constructed as follows: take the set ${\displaystyle F(B)}$ of all functions ${\displaystyle f:B\to F}$ such that ${\displaystyle f(b)=0}$ for all but finitely many ${\displaystyle b}$ in ${\displaystyle B.}$ These functions can be added and multiplied with elements of ${\displaystyle F}$ to obtain the desired ${\displaystyle F}$-vector space.

An important result about dimensions is given by the rank–nullity theorem for linear maps.

If ${\displaystyle F/K}$ is a field extension, then ${\displaystyle F}$ is in particular a vector space over ${\displaystyle K.}$ Furthermore, every ${\displaystyle F}$-vector space ${\displaystyle V}$ is also a ${\displaystyle K}$-vector space. The dimensions are related by the formula

$${\displaystyle \dim _{K}(V)=\dim _{K}(F)\dim _{F}(V).}$$

In particular, every complex vector space of dimension ${\displaystyle n}$ is a real vector space of dimension ${\displaystyle 2n.}$

Some formulae relate the dimension of a vector space with the cardinality of the base field and the cardinality of the space itself. If ${\displaystyle V}$ is a vector space over a field ${\displaystyle F}$ and if the dimension of ${\displaystyle V}$ is denoted by ${\displaystyle \dim V,}$ then:

If dim ${\displaystyle V}$ is finite then ${\displaystyle |V|=|F|^{\dim V}.}$

If dim ${\displaystyle V}$ is infinite then ${\displaystyle |V|=\max(|F|,\dim V).}$

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, 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.

See also

• Fractal dimension – Ratio providing a statistical index of complexity variation with scale
• Krull dimension – In mathematics, dimension of a ring
• Matroid rank – Maximum size of an independent set of the matroid
• Rank (linear algebra) – Dimension of the column space of a matrix
• Topological dimension – Topologically invariant definition of the dimension of a spacePages displaying short descriptions of redirect targets, also called Lebesgue covering dimension

Notes

References

Sources

• Axler, Sheldon (2015). Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer. ISBN 978-3-319-11079-0.

External links

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