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Othogonal complement of a span

  1. Aug 28, 2007 #1
    1. The problem statement, all variables and given/known data
    Show that <S>[tex]^{\bot}[/tex]=S[tex]^{\bot}[/tex]

    2. Relevant equations

    3. The attempt at a solution
    I manage to show S[tex]^{\bot}[/tex][tex]\subseteq[/tex]<S>[tex]^{\bot}[/tex].
    What about the other way round? Any way of proving without using the concept of basis?
    Last edited: Aug 28, 2007
  2. jcsd
  3. Aug 28, 2007 #2


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    You need to prove that [tex]span(S)^{\bot}[/tex] is a subset of [tex]S^{\bot}[/tex]. However, you know that [tex] S \subseteq span(S) [/tex]. So, given the latter, try to prove the former.
    Last edited: Aug 28, 2007
  4. Aug 28, 2007 #3
    This is what I come out with:

    Let y[tex]\in[/tex]<S>[tex]^{\bot}[/tex], then <y,w>=0, for all w[tex]\in[/tex]<S>.
    But S[tex]\subseteq[/tex]<S>.
    Hence for y[tex]\in[/tex]<S>[tex]^{\bot}[/tex], then <y,w>=0, for all w[tex]\in[/tex]S.
    Hence y[tex]\in[/tex]S[tex]^{\bot}[/tex].
    Hence <S>[tex]^{\bot}[/tex][tex]\subseteq[/tex]S[tex]^{\bot}[/tex].

    Is it correct? Or should I say "for some w"?
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