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Subspace problem

  1. Sep 17, 2005 #1
    Suppose U, V are proper subsets of Rn and are subspaces and U is a proper subset of V. PRove that V perp is a proper subset of U perp

    Ok SO let U ={u1, u2, ..., un}
    let V = {v1,...vn}
    let V perp = {x1,x2,..., xn}
    let U perp = {w1,...wn}
    certainly u1 . w1 = 0
    (u1 + u2 ) . (w1+w2) = 0
    cu1 . cw1 = 0

    everything till now is pretty much basically understood
    i cant find a wayu to maek a proof work... Any hints would be appreciated!!
    I realy want to be able to get this one!
  2. jcsd
  3. Sep 18, 2005 #2
    can anyone help?

    i am still stuck on this problem and i need to solve it ...

    ok so far i get this part
    if U perp is orthognal to U then U perp is orthogonal to V.
    If V perp is orthognal to V then V perp is orthogonal to U
    this V perp is orthogonal to U perp

    then Wn . Xn = 0
    now [tex] U^{\bot} \bigcap U = \{0\} [/tex]
    and [tex] U^{\bot} \bigcap V = \{0\} [/tex]
    similarly it applies for V perp
    so both U perp and V perp include elements that are perpendicular to U and V and contain everything (but zero) that is perpendicular to U and V, thus not included in U and V.
    Now how would i relate U perp and V perp to each other. Is there a theorem that says that the perpendicular space to a bigger subspace is smaller than the perpendicular subspace to the subspace contained in the bigger subspace??
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