Results 1 to 2 of 2

Math Help - prove this rank problem

  1. #1
    Banned
    Joined
    Nov 2008
    Posts
    63

    prove this rank problem

     \alpha : V\rightarrow V is an endo. of a finite dimensional vector space V. Show:

     V\geq Im(\alpha)\geq Im(\alpha^{2})\geq ....

    and

     \{0\}\leq Ker(\alpha) \leq Ker(\alpha^{2})\leq ....

    Prove:
    <br />
r(\alpha^{k})-r(\alpha^{k+1})\geq r(\alpha^{k+1})-r(\alpha^{k+2})

    <br />
r(\alpha^{k})\geq r(\alpha^{k+1})
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by silversand View Post

     \alpha : V\rightarrow V is an endo. of a finite dimensional vector space V. Show:

     V\geq Im(\alpha)\geq Im(\alpha^{2})\geq ....
    if v \in \text{Im}(\alpha^{k+1}), then v=\alpha^{k+1}(u)=\alpha^k(\alpha(u)) \in \text{Im}(\alpha^k).



     \{0\}\leq Ker(\alpha) \leq Ker(\alpha^{2})\leq ....
    if u \in \ker(\alpha^k), then \alpha^k(u)=0. thus: \alpha^{k+1}(u)=\alpha(\alpha^k(u))=0, i.e. u \in \ker(\alpha^{k+1}).



    r(\alpha^{k})-r(\alpha^{k+1})\geq r(\alpha^{k+1})-r(\alpha^{k+2})
    for any i \geq 0 let V_i=\text{Im}(\alpha^i). so r(\alpha^i)=\dim V_i. from the first part of your problem we know that V_{i+1} is a subspace of V_i. so for any k \geq 0 the map f: \frac{V_k}{V_{k+1}} \longrightarrow \frac{V_{k+1}}{V_{k+2}}

    defined by: f(u + V_{k+1})=\alpha(u) + V_{k+2} is a well-defined onto homomorphism. thus \frac{V_{k+1}}{V_{k+2}} is isomorphic to a quotient of \frac{V_k}{V_{k+1}}. thus:  \dim \left(\frac{V_{k+1}}{V_{k+2}}\right) \leq \dim \left(\frac{V_k}{V_{k+1}}\right). hence:

    \dim (V_{k+1}) - \dim(V_{k+2}) = \dim \left(\frac{V_{k+1}}{V_{k+2}}\right) \leq \dim \left(\frac{V_k}{V_{k+1}}\right) = \dim (V_k) - \dim(V_{k+1}). \ \ \Box



    r(\alpha^{k})\geq r(\alpha^{k+1})
    this is trivial from the first part of your question.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: August 20th 2010, 05:32 AM
  2. rank sum problem
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: April 12th 2010, 05:20 PM
  3. Problem of rank
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 13th 2009, 11:27 AM
  4. Short proof that rows-rank=column-rank?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: June 26th 2009, 10:02 AM
  5. rank problem.....
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: April 23rd 2009, 06:36 PM

Search Tags


/mathhelpforum @mathhelpforum