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,^[a] and all bases of a vector space have equal cardinality;^[b] 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 ${\displaystyle ,}$ 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

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

${\displaystyle \dim _{\mathbb {R} }(\mathbb {R} ^{3})=3.}$

${\displaystyle \dim _{\mathbb {R} }(\mathbb {R} ^{n})=n,}$

${\displaystyle \dim _{F}(F^{n})=n}$

${\displaystyle F.}$

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