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 $n\times 1$. 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 $V$ are

• Abelian Group under Addition
• Scalar Multiplication

Abelian Group under Addition Axioms

1. Closure: $\forall A,B\in V,A+B\in G$
2. Associativity: $\forall A,B,C\in V,(A+B)+C=A+(B+C)$
3. Identity: $\exists I\text{ s.t. }\forall A\in V,I+A=A+I=A$
4. Inverse: $\forall A\in V,A^{-1}\in V\text{ s.t. }A+A^{-1}=A^{-1}+A=I$
5. Commutativity: $\forall A,B\in V,A+B=B+A$ Scalar Multiplication Axioms
6. Closure: $\forall c\in \mathbb{R},A\in V,cV\in V$
7. Associativity: $\forall c,d\in \mathbb{R},A\in V,(cd)A=c(dA)$
8. Distributivity: $\forall c\in \mathbb{R},A,B\in V,c(A+B)=cA+cB$
9. Distributivity: $\forall c,d\in \mathbb{R},A\in V,(c+d)A=cA+dA$