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Linear algebra problem

  1. Apr 23, 2013 #1
    1. The problem statement, all variables and given/known data
    Write a selfadjoint endomorphism ## f : E^3 → E^3## such that ##ker(f ) =
    L((1, 2, 1)) ## and ## λ_1 = 1, λ_2 = 2## are eigenvalues of f


    3. The attempt at a solution

    I know ##λ_3=0## because ́##ker(f ) ≠ {(0, 0, 0)}## and ## (ker(f ))^⊥ = (V0 )^⊥ = V1 ⊕ V2 ## due to the definition of selfadjoint.
    Then, my book gives the solution:
    ##(ker(f ))^⊥ = (L((1, 2, 1))^⊥ = {(α, β, −α − 2β) | α, β ∈ R} = L((1, 0, −1), (a, b, c))##
    but i don't understand where did it get that (α, β, −α − 2β) from.
    Could you please help me? thanks in advance :)
     
  2. jcsd
  3. Apr 23, 2013 #2

    Dick

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    Science Advisor
    Homework Helper

    (α, β, −α − 2β) is the subspace orthogonal to (1,2,1). You know that the eigenvectors of a self adjoint operator corresponding to different eigenvectors are orthogonal, yes? So you'll have to pick the other two eigenvectors from that space.
     
    Last edited: Apr 23, 2013
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