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:

1. Associativity of vector addition
2. commutativity of vector addition
3. identity element of vector addition
4. 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.
5. \(a(b\mathbf{v}) = (ab)\mathbf{v}\)
6. \(1\mathbf{v} = \mathbf{v}\)
7. \(a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}\)
8. \((a+b)\mathbf{v} = a\mathbf{v} + b\mathbf{v}\)