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A LinAlg Proof Involving Orthogonal Complement

  1. May 6, 2012 #1
    1. The problem statement, all variables and given/known data

    Here is the problem and my complete answer.

    Am I OK?

    Thanks!

    http://www.d-series.org/forums/members/52170-albums1546-picture8143.jpg [Broken]


    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. May 7, 2012 #2

    micromass

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    No, it is not ok. You seem to prove that if [itex]u,v\in S^\bot[/itex], that cu+v is an element of S. But this is simply not true at all.
    Likewise, you take [itex]a,b\in S^{\bot \bot}[/itex] and you conclude that these are in [itex]S^\bot[/itex]. But this is also not true.

    In short, it is NOT true that

    [tex]S^{\bot \bot}\subseteq S^\bot \subseteq S[/tex]

    How do you prove the theorem, well you need to prove two things:

    1) [itex]S^{\bot \bot}[/itex] is a subspace.
    2) [itex]S\subseteq S^{\bot \bot}[/itex].
     
  4. May 8, 2012 #3
    Thank you!

    By the skin of my teeth, some help from you, and the grace of God, I received the best grade I could have expected in Linear Algebra.

    Thanks, again!

    Joe
     
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