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Orthogonal Sets

  1. Mar 22, 2012 #1
    Folks,

    I am looking at my notes. Wondering where the highlighted comes from.
    Prove that a finite orthogonal set is lineaarly independent

    let u=(x_1,x_2,x_n) bee an orthogonal set set of vectors in an ips.
    To show u is linearly independent suppose

    Ʃ ##\alpha_i x_i=0## for i=1 to n

    Fix any j=1 and consider <Ʃ##\alpha_i x_i, x_j##> i=1 to n

    then

    0=<Ʃ##\alpha_i x_i, x_j##> i=1 to n

    =Ʃ<##\alpha_i x_i, x_j##> i=1 to n

    =Ʃ##\alpha_i <x_i, x_j>## i=1 to n

    =##\alpha_j <x_j, x_j>## since u is an orthonormal set


    Where does this line come from? Thanks
     
  2. jcsd
  3. Mar 22, 2012 #2
    You are assuming the x_i's are orthogonal (i.e. <x_i,x_j>=0 if i =/= j), so the only term in the sum which is not necessarily 0 is a_j<x_j,x_j>
     
  4. Mar 28, 2012 #3
    where <x_j,x_j>=1 when i=j? Thanks in advance.
     
  5. Mar 28, 2012 #4

    HallsofIvy

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    Staff Emeritus
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    Not necessarily. If you were to use "orthonormal" instead of just "orthogonal", then that would be true.

    However, that is not necessary to your proof. [itex]<x_j, x_j>[/itex] is some non-zero number. [itex]a_j[/itex]. Divide both sides of [itex]\alpha_j a_j= 0[/itex] by that number to get [itex]\alpha_j= 0[/itex].
     
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