Question

So here is a theoretical question. Let L1 and L2 be linear transformation from a vector space V into Vector space W. Let {v1,v2,....,vn} be a basis for V. Show that if L1(vi)=L2(vi) for i=1,2,....,n then L1(v)=L2(v) for any v in V.

Answer 1

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)