Got a bizzare question, appreciate any hints

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In summary, the conversation discusses various properties and theorems related to projections and orthogonal projections in a finite dimensional vector space. It is shown that if P is a projection, then I-P is also a projection and that V can be expressed as the direct sum of the image of P and the kernel of P. It is also shown that in an inner product space, P is orthogonal to its kernel if and only if P is equal to its transpose conjugate. Another result is that if P and Q are orthogonal projections, then PQ is an orthogonal projection if and only if PQ = QP, and in this case, the image of PQ is the intersection of the images of P and Q. Finally, it is shown that P+Q
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: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 I am P = Ker(I-P) and that
V = the direct sum of I am P and Ker P

b. Suppose that V is also an inner product space; show that
I am 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
I am PQ = intersection of I am P and I am 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 I am P and I am Q
 
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  • #2
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.
 
  • #3
thanks matt, time to review my maths notes :redface:
 
  • #4
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.
 

FAQ: Got a bizzare question, appreciate any hints

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