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Got a bizzare question, appreciate any hints

  1. Oct 25, 2004 #1
    :bugeye: :bugeye: :bugeye:
    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
     
  2. jcsd
  3. Oct 25, 2004 #2

    matt grime

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    Well, what's bizarre?

    a is easy computation and applying a standard theorem

    these are all 'show from something satisfies a definition'; which bits don't you understand?

    b. for instance uses (Px,y)=(x,P^ty) and the definition of orthogonal.
     
  4. Oct 25, 2004 #3
    thanks matt, time to review my maths notes :redface:
     
  5. Oct 26, 2004 #4

    matt grime

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    have you met the standard theorem I allude to? it can be shown without appeal to it as well, and you should probably do that. HINT

    1 = 1-P+P

    where 1 means the identity matrix.

    have you done the computation to show 1-P is a projection? (recall a projection is a map Q such that Q^2=Q, or equally, Q(1-Q)=0. what is the characteristic polynomial of Q? what is the minimal poly of Q if Q is neither 1 nor the zero map?)

    if you can't see how to start a question it is always advisable to read the notes that tell you the defining properties of what you wish to demonstrate.
     
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