vector space

A vector space or linear space is an algebraic structure consisting of an abelian group $V$ of vectors under addition, with a field $F$ of scalars, equipped with associative scalar multiplication which distributes over addition.

Formally,
for all $\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}\in V$, and all $a,b,c\in F$,

$(V,{+}_V)$ is an abelian group:

1. $(V,{+}_V)$ is closed: $(\overrightarrow{u}+\overrightarrow{v})\in V$
2. $(V,{+}_V)$ is associative: $(\overrightarrow{u}+\overrightarrow{v})+\overrightarrow{w}=\overrightarrow{u}+(\overrightarrow{v}+\overrightarrow{w})$
3. $(V,{+}_V)$ has an identity element: $\overrightarrow{v}+{\overrightarrow{0}}_V=\overrightarrow{v}={\overrightarrow{0}}_V+\overrightarrow{v}$
4. $(V,{+}_V)$ has inverses: $\overrightarrow{v}+(-\overrightarrow{v})={\overrightarrow{0}}_V=(-\overrightarrow{v})+\overrightarrow{v}$
5. $(V,{+}_V)$ is commutative: $\overrightarrow{u}+\overrightarrow{v}=\overrightarrow{v}+\overrightarrow{u}$

$(F,{+}_F,{\times }_F)$ is a field:

6. $(F,{+}_F)$ is closed: $a+b\in F$
7. $(F,{+}_F)$ is associative: $(a+b)+c=a+(b+c)$
8. $(F,{+}_F)$ has an identity element: $a+0_F=a=0_F+a$
9. $(F,{+}_F)$ has inverses: $a+(-a)=0=(-a)+a$
10. $(F,{+}_F)$ is commutative: $a+b=b+a$
11. $(F,{\times }_F)$ is closed: $ab\in F$
12. $(F,{\times }_F)$ is associative: $(ab)c=a(bc)$
13. $(F,{\times }_F)$ has an identity element: $a{1}_F=a=1_{F}a$
14. $(F,{\times }_F)$ has inverses: $a(a^{-1})=1_F=(a^{-1})a$
15. $(F,{\times }_F)$ is commutative: $ab=ba$
16. $(F,{+}_F,{\times }_F)$ is left-distributive: $a(b+c)=ab+ac$
17. $(F,{+}_F,{\times }_F)$ is right-distributive: $(a+b)c=ac+bc$
18. Field identities are unique: $0_F\ne 1_F$

$(F,V,\otimes )$ scalar multiplication with vectors:

19. $(F,V,\otimes )$ is associative with $(F,{\times }_F)$: $a(b\overrightarrow{v})=(ab)\overrightarrow{v}$
20. $(F,V,\otimes )$ has an identity element: $1_F\overrightarrow{v}=\overrightarrow{v}$
21. $(F,V,\otimes )$ distributes over $(V,{+}_V)$: $a(\overrightarrow{u}+\overrightarrow{v})=a\overrightarrow{u}+a\overrightarrow{v}$
22. $(F,V,\otimes )$ distributes over $(F,{+}_F)$: $(a+b)\overrightarrow{v}=a\overrightarrow{v}+b\overrightarrow{v}$