Results 1 to 2 of 2

Thread: Proving an identity

  1. #1
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,656
    Thanks
    14

    Proving an identity

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

    No idea how to show this!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    Posts
    678
    Thanks
    1
    Quote Originally Posted by Krizalid View Post
    Let $\displaystyle S,W$ be subspaces of $\displaystyle V$ where $\displaystyle V$ is a vector space with inner product and $\displaystyle \dim V=n,$ show that $\displaystyle (S\cap W)^\perp=S^\perp+W^\perp.$

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

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

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

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

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

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

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

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

    Hence we are done.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proving an identity
    Posted in the Trigonometry Forum
    Replies: 5
    Last Post: Jul 17th 2011, 12:23 PM
  2. Proving Another Identity
    Posted in the Trigonometry Forum
    Replies: 7
    Last Post: Jun 6th 2011, 10:34 AM
  3. Help proving an identity
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: Apr 22nd 2010, 11:57 AM
  4. Proving an identity that's proving to be complex
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: Jul 21st 2009, 01:30 PM
  5. proving an identity
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: Mar 16th 2008, 10:03 AM

Search Tags


/mathhelpforum @mathhelpforum