Coordinate Space
Perhaps the most important example of a vector space is the following. For any positive integer n, the space of all n-tuples of elements of F forms an n-dimensional vector space over F sometimes called coordinate space and denoted Fn. An element of Fn is written
where each xi is an element of F. The operations on Fn are defined by
The most common cases are where F is the field of real numbers giving the real coordinate space Rn, or the field of complex numbers giving the complex coordinate space Cn.
The quaternions and the octonions are respectively four- and eight- dimensional vector spaces over the reals.
The vector space Fn comes with a standard basis:
where 1 denotes the multiplicative identity in F.
Read more about this topic: Examples Of Vector Spaces
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