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Let V be a finite dimensional vector space, and P <- L(V, V) be a projection, i.e P = P^2

a. Show that I - P is also a projection, that Im P = Ker(I-P) and that

V = the direct sum of Im P and Ker P

b. Suppose that V is also an inner product space; show that

Im P orthognoal to Ker P <=> P = "P transpose conjugate"

c. Show that if P, Q are orthogonal projections, then PQ is a

orthogonal projection <=> PQ = QP, and that in this case

Im PQ = intersection of Im P and Im Q

d. Show that if P, Q are orthogonal projections, then P+Q is an

orthogonal projection <=> PQ = 0, and that in this case

Im(P+Q) = direct sum of Im P and Im Q

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# Got a bizzare question, appreciate any hints

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