Say we have a transformation T[tex]\in[/tex]L(V). Now suppose a subspace of V (U) is in the rangespace of T. Now suppose PUv=u with u=a1u1+...+amvm. Now apply T to u to get T(u)=b1u1+...+bmum=/=a1u1+...+amvm. What would happen if we apply PUto T(u)? In other words, what would we end up with after computing PUT(u)? I'm just wondering whether or not applying a transformation (a non-projection in this case) after a projection would result in a different output after applying the projection to the image of T? In other words, would PUT(u) map to a1u1+..+amum or would it map to b1u1+..+bmum? Thank you for your response!