Linear combinations and basis

Hi there could anybody help me with this question maybe just showing me the first part than guiding me through the rest myself that would be so so appreciated.

Let U be a vector space with ordered basis P = [e1,e2] and let V be a vector space with ordered basis Q = [ f1, f2]. Let T : U → V be a linear map with matrix $$A = \pmatrix{2&3\\3&5}$$ with respect to the ordered bases P and Q. Suppose Q' =[f1',f2'] is another basis of V and suppose that T has matrix $$B = \pmatrix{4&6\\1&7}$$ with respect to the ordered bases P and Q'. Express the elements of Q as linear combinations of elements of Q'.

Recall that the columns of $A,B$ describe the associated transformation. For example, the second column of $A$ tells us that $$T(e_2) = 3f_1 + 5f_2$$ So, we therefore know that $$\Big(T(e_1) =\Big) 2f_1 + 3f_2 = 4f_1' + f_2'\\ \Big(T(e_2) =\Big) 3f_1 + 5f_2 = 6f_1' + 7f_2'$$ Rewrite these equations in terms of the coordinate vectors relative to the basis $Q'$. We have $$2[f_1]_{Q'} + 3[f_2]_{Q'} = \overbrace{\pmatrix{4\\1}}^{[4f_1' + f_2']_{Q'}}\\ 3[f_1]_{Q'} + 5[f_2]_{Q'} = \pmatrix{6\\7}$$ Rewriting this with matrices, we have $$\pmatrix{2&3\\3&5} \pmatrix{[f_1]_{Q'} & [f_2]_{Q'}} = \pmatrix{4&6\\1&7}$$ So, the coefficients in our desired linear combination are the columns of $$\pmatrix{[f_1]_{Q'} & [f_2]_{Q'}} = \pmatrix{2&3\\3&5}^{-1}\pmatrix{4&6\\1&7}$$