Answer :
Final answer:
The sum of orthogonal projections P and Q (P+Q) is the identity operator, while their multiplication (PQ) is the zero map. P - Q is proven to be its own inverse by showing that squaring it gives the identity operator.
Explanation:
Orthogonal projections are often involved in mathematical problems, and understanding them geometrically aids in their mathematical manipulation. P and Q are orthogonal projections onto the subspace E and the orthogonal complement E⊥, respectively.
From a geometric viewpoint, each vector in our space gets broken into two components - one in the subspace E (by P) and the other in the orthogonal complement (by Q). When you sum these operations (P + Q), you're essentially combining these two components back into the original vector, leading us to the conclusion that P + Q is the identity operator, I.
As for the operation PQ, since P projects vectors into subspace E and Q projects vectors into E⊥ which is orthogonal to E, the result of PQ applied to any vector would be a vector which belongs both to E and E⊥. The only vector that fulfills this condition is the zero vector, leading to the conclusion that PQ is the zero map.
To prove P - Q is its own inverse, we must show (P - Q)^2 is the identity operator I. With some algebraic manipulation, and using the properties we proved earlier, we find: (P - Q)^2 = P^2 - PQ - QP + Q^2 = P - 0 - 0 + Q = P + Q = I.
Learn more about Orthogonal Projections here:
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