Vector Space

An Algebraic Structure containing Tensor elements called Vectors, over a Algebraic Field of Scalar values.

Vector Space Element

Note that Vectors need not be . Any possible element of a vector space is by definition, a vector. This means that vector spaces really contain Tensors. 99% of the time when people are talking about vectors, they really mean “Coordinate vector”, i.e. a vector that represents a location in Space. Whereas a vector represents a location in a Vector Space.

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Axioms

The Axioms neccessary to qualify as a Vector Space are

• Abelian Group under Addition
• Scalar Multiplication

Abelian Group under Addition Axioms

1. Closure:
2. Associativity:
3. Identity:
4. Inverse:
5. Commutativity: Scalar Multiplication Axioms
6. Closure:
7. Associativity:
8. Distributivity:
9. Distributivity: