Is my proof valid? Let W1,W2 be subspaces finite-dim, IPS V with T1,T2 orthogonal projections on W1,W2. Show that T1T2=T0 iff W1 perp W2.

Question

Is this a good proof? I'm most concerned about the very last line.. does what I show actually imply what I say it does?

Answer 1

Yes this is valid. However, to be very clear you should say "If $\langle x, T_1 T_2 y \rangle = 0$ for every possible $x \in W_1$ for a given $y$, then $T_1T_2y$ must also be $0$. Therefore, if $T_1T_2y = 0$ for every possible $y \in W_2$, we must furthermore have $T_1T_2 = T_0$."

I think the confusion you had was going from the inner product = 0 to the RHS = 0. In general, if you have an inner product of 0 with one vector, then you are orthogonal to that vector. But if you have an inner product of 0 with every vector, then you must be orthogonal to the entire space. The only vector orthogonal to the entire space and still in that space is the 0 vector.