Vector space/Example I

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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.