Vector Space Example 1

• 1. If v,w, in V then v+w=v₁+w₁,v₂+w₂,v₃+w₃,...,v_n+w_n. But for all v_i, w_i, 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,...,u_n+ v₁+w₁,v₂+w₂,v₃+w₃,...,v_n+w_n=

          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*v_n.

          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. 

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