Vector Space Dimension

We say ''V'' is finite-dimensional if the dimension of ''V'' is . EXAMPLES E.g. The vector space R³ has {(1,0,0), (0,1,0), (0,0,1)} as a basis, and therefore we have dim_R(R³) = 3. More generally, dim_R(R^''n'') = ''n''. And more generally still, dim_''F''(''F''^''n'') = ''n''. The Complex Number s C are both a real and complex vector space; we have dim_'''R'''(C) = 2 and dim_C(C) = 1. So the dimension depends on the base field. The only vector space with dimension 0 is {0}, the vector space consisting only of its zero element. FACTS If ''W'' is a Linear Subspace of ''V'', then dim(''W'') ≤ dim(''V''). To show that two finite-dimensional vector spaces are equal, one often uses the following criterion: if ''V'' is a finite-dimensional vector space and ''W'' is a linear subspace of ''V'' with dim(''W'') = dim(''V''), then ''W'' = ''V''.
	:If Dim''V'' Is Finite, Then ''V''	''F''<sup>dim''V''</sup>
	:If Dim''V'' Is Infinite, Then ''V''	max(''F'', dim''V'')

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