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Thread: Ordered Basis Problem

  1. #1
    Junior Member
    Apr 2008

    Ordered Basis Problem

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

    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 u_1,u_2,u_3\inV such that

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

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

    My inital procedure was to first find a common eigenvector for x,y,z and take that as u_1, but I couldn't find a common eigenvector that worked, when I calculated u_2, u_3, the respective matrices for x,y,z were not stricly upper triangular. I then tried 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 09:05 AM.
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