Dimension of a vector space

Dimension of a vector space: The dimension of a vector space over a field equals the cardinality of a basis for that vector space (we can always find a basis for any vector space, using the axiom of choice, and any two bases have equal cardinality).

A field has dimension one as a vector space over itself. Dimension of a direct sum of vector spaces is the sum of their dimensions, and dimension of a tensor product of vector spaces is the product of their dimensions.

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