Results 1 to 1 of 1

Thread: Proof concerning inner products

  1. #1
    Jul 2009

    Proof concerning inner products

    Prove that dimV=dimU^$\displaystyle \perp$+dimU

    Let T project from V---->U. Assume V=nullT+rangeT.
    let dim V=n, dim rangeT=m, and dim nullT=k.
    Extend the basis of V by {, w1...wk} where wi is in nullT
    and uj is in rangeT, for 1<=i<=k and 1<=j<=m. Since T(uj)=uj and T(wi)=0, (for wi=/=0)
    <T(uj), T(wi)>=0, thus wi is 0 in rangeT, and is orthogonal to uj, by the inner
    product being 0, so wi is in both nullT and orthU. Therefore, nullT=U^$\displaystyle \perp$,
    so dim nullT=dimU^$\displaystyle \perp$=k. Since T maps from V---->U and V-nullT=rangeT,
    knowing that nullT is not in U means that rangeT=U, thus dim rangeT=dimU=m, and dimV=dimnullT+dimrangeT=dimU^$\displaystyle \perp$+dimU=m+k.

    ami rite?
    Last edited by evilpostingmong; Jul 5th 2009 at 05:53 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof that all numbers are products of primes?
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: Feb 8th 2011, 01:56 AM
  2. Replies: 3
    Last Post: Mar 30th 2010, 03:53 PM
  3. Vector Proof - Dot and Cross Products
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Feb 25th 2009, 06:05 PM
  4. proof of two cross products.
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Feb 23rd 2009, 10:26 PM
  5. Cartesian Products: Proof of Equality of Sets
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: May 9th 2006, 02:46 PM

Search Tags

/mathhelpforum @mathhelpforum