Results 1 to 2 of 2

Math Help - linear operator

  1. #1
    Newbie
    Joined
    Oct 2008
    Posts
    20

    Wink linear operator

    I posted this before but I did typo. Now should ne correct but I still can't make it.
    Please, help me:
    Let T:V->V be a linear operator on the vector space over the field F. Let v is in V and let m be a positive integer for which v is not equal to 0, T(v) is not equal to 0, ...,T^(m-1)(v) is not equal to 0, but T^(m)(v) is equal to 0. Show that {v, T(v), ... , T^(m-1)(v)} is linearly independent set.
    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by maria_stoeva View Post
    I posted this before but I did typo. Now should ne correct but I still can't make it.
    Please, help me:
    Let T:V->V be a linear operator on the vector space over the field F. Let v is in V and let m be a positive integer for which v is not equal to 0, T(v) is not equal to 0, ...,T^(m-1)(v) is not equal to 0, but T^(m)(v) is equal to 0. Show that {v, T(v), ... , T^(m-1)(v)} is linearly independent set.
    Thank you!
    suppose \sum_{j=0}^{m-1}c_jT^j(v)=0, and c_j \neq 0, for some 0 \leq j \leq m-1. choose 0 \leq k \leq m-1 to be the smallest such that c_k \neq 0. then 0=T^{m-1-k} \left(\sum_{j=0}^{m-1}c_jT^j(v) \right)=c_kT^{m-1}(v), which gives us:

    T^{m-1}(v)=0 because c_k \neq 0. that contradicts our assumption that T^{m-1}(v) \neq 0. \ \ \ \Box
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear operator
    Posted in the Differential Geometry Forum
    Replies: 11
    Last Post: June 5th 2011, 10:41 PM
  2. Linear operator
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 30th 2011, 09:43 PM
  3. Linear Operator
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 25th 2011, 11:59 PM
  4. Is this Linear Operator?
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 14th 2010, 01:59 PM
  5. linear operator??!
    Posted in the Advanced Math Topics Forum
    Replies: 5
    Last Post: June 16th 2007, 07:28 AM

Search Tags


/mathhelpforum @mathhelpforum