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Linear algebra Matrix with respect to basis

  1. Nov 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the matrix of the linear operator with respect to the given basis B.

    D: P2 -> P2 defined by D(ax2 + bx + c) = 2ax+b, B = { 3x2+2x+1, x2-2x, x2+x+1 }

    2. Relevant equations


    3. The attempt at a solution

    I set the basis B = { (3,2,1), (1,-2,0), (1,1,1) } based on the equations

    then I did
    D(3,2,1) = 6x+2 = (0,6,2)
    D(1,-2,0) = 2x-2 = (0,2,-2)
    D(1,1,1) = 2x-1 = (0,2,-1)

    I believe those are the steps I have to take? I'm just not sure where to go from here.

  2. jcsd
  3. Nov 11, 2009 #2


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    Homework Helper

    I would probably start with the D matrix in the {x^2,x,1} basis, then use the B matrix you found to transform it into the new basis
  4. Nov 11, 2009 #3


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    now for clarity let the bases be given by
    [tex] e = {e_1, e_2, e_3} = {x^2,x,1}[/tex]

    [tex] b = {b_1, b_2, b_3} = { 3x2+2x+1, x2-2x, x2+x+1 } [/tex]

    so as another way, you have the effect of the matrix D on the b basis vectors, you could re-write the resultant vectors in terms of the b basis to find D relative to b (call it Db)

    though the first method outlined is probably better
  5. Nov 11, 2009 #4
    Hmm, I don't know if I quite follow. So:

    D(1,0,0) = 2x = (0,2,0)
    D(0,1,0) = 1 = (0,0,1)
    D(0,0,1) = 0 = (0,0,0)

    So what would be the next step?
  6. Nov 11, 2009 #5


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    ok, so given a column vector in the e basis ue = (a,b,c)T representing (ax^2+b+c), what is the matrix De, such that result is ve in also the e basis

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

    you've got all the values, just put them in matrix form
    De =
    [ ? ? ?]
    [ ? ? ?]
    [ ? ? ?]

    the you need to find the matrix T, that transforms from the b basis to the v basis, so given a vector ub, what matrix T, takes it to ue

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

    ub & ue are the same vector just written in different bases. The matrix T will be very closely related to the set of vectors B you gave.
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