Results 1 to 2 of 2

Math Help - Nilpotent

  1. #1
    Junior Member
    Joined
    Aug 2009
    Posts
    62

    Nilpotent

    Let T:V-->V be a nilpotent linear map of index =Dim(V)=n.Then show that dim(Ker(T))=1.

    We know that {v,Tv,...,T^n-1(v)} is linearly independent in V.now how do i show further..
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by math.dj View Post
    Let T:V-->V be a nilpotent linear map of index =Dim(V)=n.Then show that dim(Ker(T))=1.

    We know that {v,Tv,...,T^n-1(v)} is linearly independent in V.now how do i show further..

    You mean: let be 0\neq v\in V s.t. Tv,T^2v,\ldots,T^{n-1}v\neq 0 , then \{v,Tv,\ldots,T^{n-1}v\} are lin. ind. and thus a basis of V...well, you're done, since then \{Tv,\dots,T^{n-1}v\} are lin. ind. AND contained in Im(T) , so now just use the dimensions theorem...

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Nilpotent
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 29th 2011, 04:41 PM
  2. Nilpotent
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: April 18th 2010, 09:11 AM
  3. nilpotent
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 6th 2009, 06:34 PM
  4. nilpotent
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 24th 2009, 10:49 AM
  5. Nilpotent Matrix
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 29th 2008, 01:26 AM

Search Tags


/mathhelpforum @mathhelpforum