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Change of basis Matrix

  1. Nov 1, 2009 #1
    1. The problem statement, all variables and given/known data


    Let {e1,e2,e3} be a basis for the vector space V, and [itex]T:V \rightarrow V [/itex] a linear transformation.

    let f1 ;= e1 f2;=e1+e2 f3;=e1+e2+e3

    Find the Matrix B of T with respect to {f1,f2,f3} given that the matrix with respect to {e1,e2,e3} is

    [itex]\[ \left( \begin{array}{ccc}1 & 1 &1 \\1 & 1 & 0\\ 1 & 0 & 0\\\end{array} \right)^{-1}\] [/itex]

    2. Relevant equations



    3. The attempt at a solution


    Let A be the matrax of the basis {f1,f2,f3}
    A= [itex]\[ \left( \begin{array}{ccc}1 & 1 &1 \\0 & 1 & 1\\ 0 & 0 & 1\\\end{array} \right)\] [/itex]
    Than,

    B= [itex]\[ \left( \begin{array}{ccc}1 & 1 &1 \\0 & 1 & 1\\ 0 & 0 & 1\\\end{array} \right)^{-1}\] [/itex][itex]\[ \left( \begin{array}{ccc}1 & 1 &1 \\1 & 1 & 0\\ 1 & 0 & 0\\\end{array} \right)\] [/itex][itex]\[ \left( \begin{array}{ccc}1 & 1 &1 \\0 & 1 & 1\\ 0 & 0 & 1\\\end{array} \right)\] [/itex]


    B=[itex]\[ \left( \begin{array}{ccc}0 & 0 &1 \\0 & 1 & 1\\ 1 & 1 & 1\\\end{array} \right)\] [/itex]


    Does this look allright?
     
  2. jcsd
  3. Nov 2, 2009 #2

    lanedance

    User Avatar
    Homework Helper

    comments as below, tried to follow your work
    there is an inverse sign, that you don't seem to carry later on

    so based on the question we have the linear transform, in the e basis given by say:

    [tex] u^e = T^e.v^e[/tex]

    with
    [tex] T^e=[/tex]
    [tex]\[ \left( \begin{array}{ccc}1 & 1 &1 \\1 & 1 & 0\\ 1 & 0 & 0\\\end{array} \right)^{-1}\] [/tex]
    looks ok here, based on the defintion of the basis, the matrix A transforms a vector from the f basis to the e basis, so
    [tex] u^e = A.u^f [/tex]

    and using the inverse
    [tex] u^f = A^{-1}.u^e [/tex]

    now looking at the transformation relation
    [tex] u^f = A^{-1}u^e = A^{-1}T^e.v^e = A^{-1}T^e(Av^f) = (A^{-1}T^eA)v^f[/tex]

    the matrix B in the f basis is
    [tex] u^f = (A^{-1}T^eA)v^f = B v^f[/tex]

    now unless i'm following wrong, it looks like you've dropped an inverse sign, and i'm not too sure how you simplify to get to the next line in any case...
     
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