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 ${\displaystyle V}$ 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.

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.

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 ${\displaystyle W=V.}$