So here is a theoretical question. let l1 and l2 be linear transformation from a vector space v into vector space w. let {v1,} be a basis for v. show that if l1(vi)=l2(vi) for i=1, then l1(v)=l2(v) for any v in v.

if v belongs to V, then we can find scalars a1,a2,...,an, such that v=a1*v1+a2*v2+...+an*vn, L1(v)=L1(a1*v1+a2*v2+...+an*vn) =a1*L1(v1)+a2*L1(v2)+...+an*L1(vn) =a1*L2(v1)+a2*L2(v2)+...+an*L2(vn) =L2(a1*v1+a2*v2+...+an*vn) =L2(v)

The answer is the third choice