Results 1 to 2 of 2

Math Help - Linear Algebra. Maps!!

  1. #1
    Newbie
    Joined
    Apr 2008
    Posts
    11

    Linear Algebra. Maps!!

    Let T and T* be linear mappings on R^n defined by T(x)=Ax, and T*(x)=(A^T)x (A^T = A transpose). Show that kerT=(range of T*)perpendicular (ie, not the range of T*, but the set of vectors which are perpendicular to the range of T*. THANKS TO ANYONE WHO CAN HELP ME OUT WITH THIS EVEN THE SLIGHTEST!!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by flawless View Post
    Let T and T* be linear mappings on R^n defined by T(x)=Ax, and T*(x)=(A^T)x (A^T = A transpose). Show that kerT=(range of T*)perpendicular (ie, not the range of T*, but the set of vectors which are perpendicular to the range of T*. THANKS TO ANYONE WHO CAN HELP ME OUT WITH THIS EVEN THE SLIGHTEST!!
    The key fact is that \langle Tx,y\rangle = \langle x, T^*y\rangle for all x,y∈R^n, where the angled brackets denote the inner product. In matrix notation, the same equation can be written y^{\textsc t}Ax = x^{\textsc t}A^{\textsc t}y.

    In that equation, it's kind of obvious that Tx=0 if and only if x is perpendicular to everything in the range of T*.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear maps
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: October 9th 2010, 03:42 AM
  2. Linear Maps
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 15th 2009, 09:27 AM
  3. Linear maps
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 27th 2008, 03:25 AM
  4. Linear maps. Proving its linear, and describing kernal.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 20th 2008, 12:46 AM
  5. linear maps
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: March 9th 2008, 09:01 PM

Search Tags


/mathhelpforum @mathhelpforum