Example I:
Let V be the set of all n-tuples, [v1,v2,v3,...,vn] where vi, for i={1,2,3,...n} is a member of R={real numbers}. Let the field be R, as well.
Define Vector Addition:
For all v, w, in V, define v+w=[v1+w1,v2+w2,v3+w3,...,vn+wn].
Define Scalar Multiplication:
For all a in F and v in V, a*v=[a*v1,a*v2,a*v3,...,a*vn]. Then V is a Vector Space over R.
Proof:
- 1. If v,w, in V then v+w=[v1+w1,v2+w2,v3+w3,...,vn+wn]. But for all vi, wi, where
i={1,2,3,...,n}, vi+wi is in R, since R is a field. Therefore, for all u,v in V, v+w is in V.
- 2. If u,v,w in V then u+(v+w)= [u1,u2,u3,...,un]+ [v1+w1,v2+w2,v3+w3,...,vn+wn]=
[u1+(v1+w1),u2+(v2+w2),u3+(v3+w3),...,un+(vn+wn)]. But for all ui,vi,wi, where i={1,2,3,...,n}, ui+(vi+wi)=(ui+vi)+wi, since ui,vi,wi in R and R is a field. Therefore, u+(v+w)=(u+v)+w, for all u,v,w in V.
- 3. Since R is a field there exists an additive identity in R, say, 0. Consider
[0,0,0,...,0]. Then 0 is in V. But then for all v in V, 0+v= [v1+0,v2+0,v3+0,...,vn+0]= [v1,v2,v3,...,vn] since vi in R for all i={1,2,3,...,n}, and 0+vi=vi for all vi where i={1,2,3,...,n}, since R is a field.
- 4. Since R is a field, there exists for every a in R and element -a in R such that
a+(-a)=0. For v in V=[v1,v2,v3,...,vn], Consider -v=[-v1,-v2,-v3,...,-vn]. -v is in R and v+(-v)=[v1+(-v1),v2+(v2),v3+v3+(-v3),...,v+(-vn)]=0, since vi+(-vi)=0 for all i={1,2,3,..,n} since R is a field.
- 5. Since R is a field, for a,b in R a+b=b+a. Then
v+w=[v1+w1,v2+w2,v3+w3,...,vn+wn]= [w1+v1,w2+v2,w3+v3,...,wn+vn] =w+v, since for each i={1,2,3,...,n} vi+wi=wi+vi, since R is a field.
- 6. Since R is a field, if a,b in R a*b in R. Then a*v=[a*v1,a*v2,a*v3,...,a*vn].
Then a*vi for i={1,2,3,...,n} is in R. Therefore, a*v in V.
- 7. Since R is a field, R has a multiplicative identity 1, such that 1*a=a for all
a in R. Then for v in V, 1*v=[1*v1,1*v2,1*v3,...,1*vn]= [v1,v2,v3,...,vn]=v, since for all vi, for i={1,2,3,...,n}, 1*vi=*vi.
- 8. Since R is a field for a,b,c in R a*(b+c)=a*b+a*c. Then for v in V
a*(v+w)=a*[v1+w1,v2+w2,v3+w3,...,vn+wn]= [a*(v1+w1),a*(v2+w2),a*(v3+w3),...,a*(vn+wn)]= [a*v1+aw1,a*v2+a*w2,a*v3+a*w3,...,a*vn+a*wn]= a*[v1,v2,v3,...,vn]+a*[w1,w2,w3,...,,wn]=a*v+a*w.
- 9. Since R is a field, for a,b,c in R a*(b*c)=(a*b)*c.
Then a*(b*v)=a*[b*v1,b*v2,b*v3,...,b*vn]= [(a*b)v1,(a*b)v2,(a*b)v3,...,(a*b)vn]=(a*b)*v.
- 10. Since R is a field, for a,b,c in R, (a+b)*c=a*b+a*c.
Then (a+b)v=(a+b)[v1,v2,v3,...,vn]=[(a+b)v1,(a+b)v2,(a+b)v3,...,(a+b)vn]= [a*v+b*v1,a*v2+a*v2,a*v3+b*v3,...,a*vn+b*n]=[a*v1,a*v2,a*v3,...,a*vn]+ [b*v1,b*v2,b*v3,...,b*vn]=a*v+b*v.
This vector space is denoted Rn.
- See also : Vector space