Results 1 to 1 of 1

Thread: Ordered Basis Problem

  1. #1
    Junior Member
    Joined
    Apr 2008
    Posts
    48

    Ordered Basis Problem

    Let $\displaystyle V$ be a 3-D space over a field $\displaystyle k$ with basis $\displaystyle B = (v_1,v_2,v_3)$ and consider linear operators $\displaystyle x,y,z:V\rightarrow V$ whose matrices relative to $\displaystyle B$ are:

    $\displaystyle X=\left(\begin{array}{ccc}0&1&0\\0&0&0\\0&1&0\end{ array}\right), Y=\left(\begin{array}{ccc}0&2&0\\1&0&-1\\0&2&0\end{array}\right), Z=\left(\begin{array}{ccc}1&0&-1\\0&0&0\\1&0&-1\end{array}\right)$

    Find vectors $\displaystyle u_1,u_2,u_3\inV$ such that

    $\displaystyle x(u_1)=y(u_1)=z(u_1)=0$
    $\displaystyle x(u_2),y(u_2),z(u_2)\in \ V_1=ku_1$ with $\displaystyle u_2\notin \ V_1$
    $\displaystyle x(u_3),y(u_3),z(u_3)\in \ V_2=ku_1\oplus \ ku_2$ with $\displaystyle u_3\notin V_2$

    I have been asked to find these vectors so that the matrices of $\displaystyle x,y,z$ with respect to $\displaystyle u_1,u_2,u_3$ are stricly upper triangular.

    My inital procedure was to first find a common eigenvector for $\displaystyle x,y,z$ and take that as $\displaystyle u_1$, but I couldn't find a common eigenvector that worked, when I calculated $\displaystyle u_2, u_3$, the respective matrices for $\displaystyle x,y,z$ were not stricly upper triangular. I then tried $\displaystyle u_1=\left(\begin{array}{c}1\\0\\1\end{array}\right )$ but that didn't work. Any help would greatly appreciated.
    Last edited by skamoni; Apr 1st 2011 at 08:05 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: Aug 30th 2011, 04:48 PM
  2. Replies: 1
    Last Post: Jan 27th 2011, 07:47 PM
  3. equivalence relation of ordered basis
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Jul 17th 2010, 10:59 AM
  4. Replies: 2
    Last Post: May 7th 2010, 08:51 AM
  5. Determinant of ordered basis matrix
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Mar 20th 2008, 09:08 AM

Search Tags


/mathhelpforum @mathhelpforum