Vector Space

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Definition of Vector Space Structures or Systems

Vector space is a non-empty set $V$, of objects called vectors, which are defined by two operations, called addition and multiplication by scalars, which satisfy the 10 axioms listed below. The axioms must hold for all vectors $u,v,w\in V$ , and scalars $c,d\in \mathbb{R}$.

1. $u+v\in V$
2. $u+v=v+u$
3. $(u+v)+w=u+(v+w)$
4. $\exists u\in V:u+0=u$
5. $\forall u\in V,\exists (-u)\in V:u+(-u)=0$
6. $cu\in V$
7. $c(u+v)=cu+cv$
8. $(c+d)u=cu+du$
9. $c(du)=(cd)u$
10. $1u=u$

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By extension of the above axioms:

1. $0u=0$
2. $c0=0$
3. $-u=(-1)u$

Topics in Vector Space.

1. n-Dimensional Space → Defined a Set for the Vector Space Structures or Systems
2. Operations on V_n → Mathematical Operations for Vector Space
3. Algebraic Properties on V_n → Mathematical Properties for Vector Space
4. Components of Vector Space

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