= Dimension
{disambiguate=vector space}
{title2=$\dim V$}
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= Dimension of a vector space
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= Dimensions of a vector space
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= Dimension
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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>, $\mathbb R[x]_{\leq2}$ 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.
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