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Math Help - projective space

  1. #1
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    projective space

    Regard the n-dimensional real projective space RPn as the
    space of lines in Rn+1 through {0}, i.e.
    RPn = (Rn+1 − {0})=∼ with x ∼ y if y = landa x for landa ̸= 0 ∈ R ;
    with the equivalence class of x denoted by [x].

    (i) Work out the necessary and sufficient condition on a linear map
    f : Rn+1 → Rm+1 for the formula [f][x] = [f(x)] to define a map
    [f] : RPn → RPm ; [x] → [f(x)] :

    (ii) For a linear map f : Rn+1 → Rn+1 satisfying the condition of (i)
    prove that the fixed point set
    Fix([f]) = {[x] ∈ RPn | [x] = [f(x)] ∈ RPn}
    consists of the equivalence classes of the lines in Rn+1 through {0} which contain eigenvectors of f.

    (iii) Construct examples of linear maps f : R3 → R3 satisfying the
    condition of (i) such that
    (a) Fix([f]) is a point.
    (b) Fix([f]) is the disjoint union of a point and a circle. [1 Mark]
    (c) Fix([f]) is a projective plane.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Two things. A) this is hard to read and I doubt most users (myself included) would want to read it. Ask separate questions is separate threads.

    Quote Originally Posted by mathshelp06 View Post
    (b) Fix([f]) is the disjoint union of a point and a circle. [1 Mark]

    B) It looks like you forgot to erase
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  3. #3
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    Quote Originally Posted by mathshelp06 View Post
    Regard the n-dimensional real projective space RPn as the
    space of lines in Rn+1 through {0}, i.e.
    RPn = (Rn+1 − {0})=∼ with x ∼ y if y = landa x for landa ̸= 0 ∈ R ;
    with the equivalence class of x denoted by [x].

    (i) Work out the necessary and sufficient condition on a linear map
    f : Rn+1 → Rm+1 for the formula [f][x] = [f(x)] to define a map
    [f] : RPn → RPm ; [x] → [f(x)] :

    (ii) For a linear map f : Rn+1 → Rn+1 satisfying the condition of (i)
    prove that the fixed point set
    Fix([f]) = {[x] ∈ RPn | [x] = [f(x)] ∈ RPn}
    consists of the equivalence classes of the lines in Rn+1 through {0} which contain eigenvectors of f.

    (iii) Construct examples of linear maps f : R3 → R3 satisfying the
    condition of (i) such that
    (a) Fix([f]) is a point.
    (b) Fix([f]) is the disjoint union of a point and a circle. [1 Mark]
    (c) Fix([f]) is a projective plane.
    This looks like part of an assessment that counts towards your final grade (my red highlighting). MHF policy is not to knowingly help with such questions. Thread closed.
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