Given a field F, a vector space over F is a set V with two operations defined on it:
The operations are required to satisfy the following axioms. For vectors u,v∈V and scalars b,c∈F:
- Addition is commutative: u+v = v+u and associative: u+(v+w) = (u+v)+w.
- There exists a zero vector 0∈V such that for all v∈V, 0+v=v and for 0*v=0.
- Multiplication is associative: b*(c*v) = (b*c)*v.
- Multiplication is distributive across addition: c*(u+v) = c*u+c*v, and (b+c)*v = b*v+c*v.
- Multiplication preserves the field's unit: 1*v = v.
Examples
- If I is any set (called an index set) and F is a field, the set of functions I→F is a vector space over F:
- Rn is a vector space over the real numbers R (take I={1,2,...,n}).
- Fn is a vector space over F.
- The set of all functions X→R is a vector space over R, for any set X.
The set of functions from R to any vector space V over R is a vector space over R. (Of course this remains true if you replace "R" with any other field "F").
The set of all continuous / differentiable / k times differentiable / analytic / almost any "nice" property functions X→Rn, where X⊆Rm is some open set is a vector space over R.
If F is a field and K⊆F is a subfield, then F is a vector space over K:
Many thanks to jrn and to scibtag for stubbornly insisting I have an error in the definitions -- the axioms (1*v=v) and (0+v=v) were missing!