Dimension (vector space)

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 ${\displaystyle [V:F],}$ read "dimension of ${\displaystyle V}$ over ${\displaystyle F}$". When ${\displaystyle F}$ can be inferred from context, ${\displaystyle \dim(V)}$ is typically written.

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