Let V be a nitedimensional Fvector space Assume that S T V

Let V be a nite-dimensional F-vector space. Assume that S, T : V V are operators such that range(S) range(T). Prove, or give a counterexample: there is an operator A on V such that S = TA.

Solution

CLAIM: There does exist a linear operator A: V->V such that S=TA

Proof: Let {v[1],v[2],...v[n]} be a basis of V.

Let w[i]= S(v[i]) , i=1,2,..n

Now w[i] =S(v[i]) belongs to the range of T (given)

So there exists a vector , say z[i] in V such that

                                          T(z[i])= w[i]=S(v[i])...............................(1)

Define a linear operator by setting

                                          A(v[i])= z[i]............................................(2)

and extending it linearly to V (this is possible, as v[i] is a basis and z[i] exists for all i=1,2...n)

From (1) and (2)              S(v[i]) = w[i]=T(z[i]) =T(A(v[i]).......................(3)

By linearity, if follows that

                                               S=TA on V..................................................(4)