A vector space V is a set of vectors over a Field F with a list of Axioms which are true.
There are two binary operations, Vector Addition (+) and Scalar Multiplication (*) which take all possible values from F and the following Axioms (a.k.a. Identities) are true:
- Closure to Addition — the sum of any two vectors is a vector.
- Closure to Multiplication — the product of any two vectors is a vector.
- Associativity of Addition — (a+b)+c=a+(b+c)
- Associativity of Multiplication — (a*b)*c = a*(b*c)
- Commutativity of Addition — a+b=b+a
- Commutativity of Multiplication — a*b=b*a
- Addition Identity Element — Zero Vector
- Multiplication Identity Element — One
- Additive Inverse — a + (-a) = 0
- Multiplicative Inverse — a * (1/a) = 0
- Distribution of Multiplication over Addition — a(b+c) = ab + bc
We might playfully imagine the axioms to be constraints and that vectors turned loose will do anything and everything they can within these constraints. We’ve also said that a vector space is built by all possible vectors doing everything vectors do. This method of building was used to construct all positive counting numbers by saying that 1 was a counting number and that the existence of counting number n gives us counting number n+1.
The Geometric Algebra of a Vector Space is an algebra over a field with its multiplication operation being the Geometric Product on a space of Multivectors containing the scalars, F, and the vector space V.
Wow. Has any other sentence screamed more loudly, “Please diagram me!”, to the reader?
Rob Sterling 