Are these transformations linear? If so, are they isomorphisms?
1) T(M) = P M P^(-1), where P = [2 3; 5 7], from R^(2x2) to R^(2x2)
2) T(f(t)) = [f(7); f(11)] from P2 to R^2
What have you tried? The definition of "linear transformation" is, of course, that f(au+ bv)= af(u)+ bf(v) for any scalars a and b and vectors u and v. Here the "vectors" are two by two matrices.
Is P(aM+ bN)P^(-1)= aPMP^(-1)+ bPNP^{-1}?
Is T(af(t)+ bg(t))= aT(f(t))+ bT(g(t))?
A linear transformation is an isomorphism if and only if it is one-to-one.
If T(M)= T(N), does it follow that M= N?
If T(f(t))= T(g(t)), does it follow that f= g?