Vector space

Definition

A vector space $V$ over a field $F$ is a non empty set $V$ and the binary operations:

• $$+:V\times V\rightarrow V$$ given by $$+(x,y)=x+y$$ - vector addition
• $$\times:F\times V\rightarrow V$$ given by $$\times(\lambda,x)=\lambda x$$ - scalar multiplication

Such that the following 8 "axioms of a vector space" hold

Axioms of a vector space

1. $$(x+y)+z=x+(y+z)\ \forall x,y,z\in V$$
2. $$x+y=y+x\ \forall x,y\in V$$
3. $$\exists e_a\in V\forall x\in V:x+e_a=x$$ - this $$e_a$$ is denoted $$0$$ once proved unique.
4. $$\forall x\in V\ \exists y\in V:x+y=e_a$$ - this $$y$$ is denoted $$-x$$ once proved unique.
5. $$\lambda(x+y)=\lambda x+\lambda y\ \forall\lambda\in F,\ x,y\in V$$
6. $$(\lambda+\mu)x = \lambda x+\mu x\ \forall\lambda,\mu\in F,\ x\in V$$
7. $$\lambda(\mu x)=(\lambda\mu)x\ \forall\lambda,\mu\in F,\ x\in V$$
8. $$\exists e_m\in F\forall x\in V:e_m x = x$$ - this $$e_m$$ is denoted $$1$$ once proved unique.

Notation

We denote a vector space as "Let $$(V,F)$$ be a vector space" often we will write simply "let $$V$$ be a vector space" if it is understood what the field is, because mathematicians are lazy

Example

Take $$\mathbb{R}^n$$ , an entry $$v\in\mathbb{R}^n$$ may be denoted $$(v_1,...,v_n)=v$$ , scalar multiplication and addition are defined as follows:

• $$\lambda\in\mathbb{R},v\in\mathbb{R}^n$$ we define scalar multiplication $$\lambda v=(\lambda v_1,...,\lambda v_n)$$
• $$u,v\in\mathbb{R}^n$$ - we define addition as $$u+v=(u_1+v_1,...,u_n+v_n)$$

Important concepts

• Linear maps - the homomorphisms and isomorphisms of vector spaces
• Linear dependence and independence
• Span, basis and dimension