Examples of vector spaces: Difference between revisions

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==Field extensions==

Suppose '''K''' is a [[subfield]] of '''F''' (cf. [[field extension]]). Then '''F''' can be regarded as a vector space over '''K''' by restricting scalar multiplication to elements in '''K''' (vector addition is defined as normal). The dimension of this vector space is called the ''degree'' of the extension. For example the [[complex number]]s '''C''' form a two -dimensional vector space over the real numbers '''R'''. Likewise, the [[real numbers]] '''R''' form an (uncountably) infinite-dimensional vector space over the [[rational number]]s '''Q'''.

If ''V'' is a vector space over '''F''' it may also be regarded as vector space over '''K'''. The dimensions are related by the formula