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 numbers 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 numbers 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
- dimKV = (dimFV)(dimKF)
For example Cn, regarded as a vector space over the reals, has dimension 2n.
Read more about this topic: Examples Of Vector Spaces
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