# Vector space

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## Definition

A vector space [ilmath]V[/ilmath] over a field [ilmath]F[/ilmath] is a non empty set [ilmath]V[/ilmath] 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.

### 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 $u+v=(u_1+v_1,...,u_n+v_n)$

## Homomorphism between vector spaces

A homomorphism between vector spaces is a linear map