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Euclidian space en U is a linear subspace

  1. Sep 22, 2012 #1
    1. The problem statement, all variables and given/known data

    show that (U[itex]\bot[/itex])[itex]\bot[/itex]=U, if (ℝ,V,+,[,.,]) an Euclidian space en U is a linear subspace of V.

    2. Relevant equations



    3. The attempt at a solution

    suppose [itex]\beta={u_1,...,u_k}[/itex] is an orthonormal basis of U.
    pick u in U. Then [itex]u=x_1u_1+...+x_ku_k[/itex] for certain x_1,...,x_k in ℝ.

    pick u'_1,...,u'_k as orthonormal basisvectors of U[itex]\bot[/itex], where u'_i[itex]\bot[/itex]u_i for all i. and consequently take basisvectors u''_1,...,u''_k for U[itex]\bot\bot[/itex] that are orthogonal to the previous. then it follows that u''=x_1u''_1+...+x_ku''_k = x_1u_1+...+x_ku_k = u.
    this means that U[itex]\subset[/itex]U[itex]\bot\bot[/itex] and also that both have the same dimensions, and consequently that they are the same subspaces (since [itex]span(u_1,...,u_k)=span(u''_1,...,u''_k)=ℝ^k [/itex]).
     
    Last edited: Sep 22, 2012
  2. jcsd
  3. Sep 22, 2012 #2

    HallsofIvy

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    Re: U[itex]\bot[/itex][itex]\bot[/itex]=U

    I am confused as to what you mean by U⊥. I would expect that to mean "the orthogonal complement of U" but that requires that U be a subset of some larger space. If you mean "orthogonal complement, what larger space are you assuming?
     
  4. Sep 22, 2012 #3
    Re: U[itex]\bot[/itex][itex]\bot[/itex]=U

    see the problem statement: [itex]U\bot[/itex] is indeed the orthogonal complement and V is the Euclidian space of which U is a subspace. [itex]U\bot\bot[/itex] is thus the orthogonal complement of the orthogonla complement.
     
    Last edited: Sep 22, 2012
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