# Dimension (vector space)

## Number of vectors in any basis of the vector space / From Wikipedia, the free encyclopedia

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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 $V$ is **finite-dimensional** if the dimension of $V$ is finite, and **infinite-dimensional** if its dimension is infinite.

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