Define T: F^2 --> P_1(F) by T(a, b) = a + bx (with P_1 denoting P sub 1) I usually prove problems such as this by constructing a matrix of T using bases for the vector spaces and then proving that the matrix is invertible, but is the following also a viable proof that T is an isomorphism? I know it is not finished, but is it a step in the right direction? let T(z) = m + nx (T(z) is contained in P_1(F)) so z = (m, n) (z is an element of F^2) this means that that the general form for all elements in P_1(F) has a pre-image in F^2, which means that T is onto(?), so therefore T is invertible and F^2 is isomorphic to P_1(F). Is this any start at all? Any suggestions?