1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    None.

    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.

    Thanks!
     
  2. jcsd
  3. Nov 11, 2009 #2

    lanedance

    User Avatar
    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

    lanedance

    User Avatar
    Homework Helper

    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

    lanedance

    User Avatar
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook