# vector space

A vector space over a field \(F\) is a set \(V\) with these two operations

- \(\mathbf{v} + \mathbf{u}\) vector addition
- \(a\mathbf{v}\) scalar multiplication

which must satisfy these axioms:

- Associativity of vector addition
- commutativity of vector addition
- identity element of vector addition
- inverse elements of vector addition. Note that the first 4 axioms say that the vector space must must form an abelian group (algebra) under vector addition.
- \(a(b\mathbf{v}) = (ab)\mathbf{v}\)
- \(1\mathbf{v} = \mathbf{v}\)
- \(a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}\)
- \((a+b)\mathbf{v} = a\mathbf{v} + b\mathbf{v}\)