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Math Help - Proving an identity

  1. #1
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    Krizalid's Avatar
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    Proving an identity

    Let S,W be subspaces of V where V is a vector space with inner product and \dim V=n, show that (S\cap W)^\perp=S^\perp+W^\perp.

    No idea how to show this!
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  2. #2
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    Quote Originally Posted by Krizalid View Post
    Let S,W be subspaces of V where V is a vector space with inner product and \dim V=n, show that (S\cap W)^\perp=S^\perp+W^\perp.

    No idea how to show this!
    Hi-Outlines of the proof.

    1. S^\perp+W^\perp \subseteq (S\cap W)^\perp
    This is easy to show.

    2. So if we show that dimension of LHS = dimension of RHS we are done

    3. S^\perp \cap W^\perp = (S + W)^\perp
    This can again be shown easily

    4. dim( W) + dim( W^\perp) = dim( V) = n
    We will use this result directly. This is applicable for any sub-space W in V.

    5. dim( S+W) = dim( S) + dim( W) - dim( S \cap W)
    We will use this result directly. This is applicable for any sub-paces S,W in V.

    Let dim( S) = s, dim( W)=w, dim( S \cap W) = i, dim( S+W) = u
    so, s+w = u+i

    dim( S^\perp+W^\perp) = dim( S^\perp)+dim( W^\perp) - dim( S^\perp \cap W^\perp)
    = (n-s)+(n-w)-(n-u)
    = (n-i)
    = dim( (S\cap W)^\perp)

    Hence we are done.
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