Proving that two subspace are orthogonal by the fact that the projection transformation satisfies: $P = P*$

Question

Let $V$ be a unitary space with finite dimentions.

Let $L_1, L_2$ be subspaces of $V$ such that $V = L_1 \oplus L_2$.

Let $P$ be the projection transformation on $L_1$ parallel to $L_2$.

Prove that if $P* = P$ so $L_1 = L_2^{\perp}$

I tried:

Now define 2 general vectors: $$ v,v' \in V $$

They can be represented as:

$$ v = l_1 + l_2 $$ $$ v' = l'_1 + l'_2 $$

For some:

$$ l_1,l'_1 \in L_1 $$

$$ l_2,l'_2 \in L2 $$

Now if $P*=P$ it holds that:

$$ <Pv,v'> = <v,Pv'> = <l_1, l'_1+l'_2> = <l_1+l_2, l'_1> $$

Opening both sides:

$$ <l_1,l'_1> + <l_1,l'_2> = <l_1,l'_1> + <l_2,l'_1> $$

Therefore we got:

$$ <l_1,l'_2> = <l_2,l'_1> $$

From that, i want somehow to conclude that both sides equal to zero, therefore $L_1 = L_2^{\perp}$

Not sure how to continue

I would like a hint - those are homework.

Thanks for all the answers.

Answer 1

Ok so the answer is just by taking:

$$ l_1 \in L_1, l_2 \in L_2 $$

And then do:

$$ <Pl_1,l_2> = <l_1,Pl_2> $$

Remembering:

$$ Pl_1 = l_1, Pl_2 = 0. $$