 Dimension (vector space)

The dimension of a vector space is the cardinality of any of its bases. In finite dimension this counts the scalars needed to specify an arbitrary vector once an ordered basis is chosen. The exchange argument behind the basis extension theorem shows that any two finite bases have the same size: an independent family cannot be larger than a spanning family.

Thus ${\mathbb{R}}^2$ has dimension two even with an oblique basis, $\mathbb{R}[x{]}_{\le 2}$ has dimension three, and the homogeneous plane in vector subspace has dimension two. This count concerns independent freedom, so an embedded plane remains two-dimensional while sitting in a three-dimensional ambient vector space. Dimension also depends on the field, as the two interpretations of $\mathbb{C}$ in vector space illustrate.

In the context of vector spaces in linear algebra, the **dimension** of a vector space is defined as the number of vectors in a basis of that vector space. A basis is a set of vectors that is both linearly independent and spans the vector space.