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Linear transformations + writing of output matrix

  1. Dec 26, 2011 #1
    1. The problem statement, all variables and given/known data

    Given the following defined transformation

    [itex]T(a + bt+ct^{2}) = (a+c) - (c+b)t + (a+b+c)t^{2} [/itex]

    find the matrix with respect to the standard basis


    From my understanding, the standard basis for a 3 element vector would
    be

    [itex](0,0,1)^{T} (0,1,0)^{T} (1,0,0)^{T}[/itex]

    2. Relevant equations

    T(u+v)= T(u) + T(v)

    λT(v) = T(λv)

    3. The attempt at a solution


    okay, if I used the defined transformation, I get the following when I put any of the standard basis into the transformation

    [itex]T(0,0,1)^{T}=1 - t + t^{2}



    T(0,1,0)^{T} = 0 -t + t^{2}



    T(1,0,0)^{T} = 1 - 0 + t^{2}

    [/itex]

    If I am correct, the matrix should be the following

    1 -1 1
    0 -1 1
    1 0 1

    However, the tutorial answers have it in the form

    1 0 1

    -1 -1 0

    1 1 1

    Shouldn't my answer be correct, since the [itex]t[/itex] and [itex]t^{2}[/itex] terms are different parts of a linear equation which is why they can't be in the same column ?
     
  2. jcsd
  3. Dec 26, 2011 #2
    T(x) = A.x

    where
    x = [c b a]^T
    A is the transformation matrix


    The way you have put your A matrix, we would need to do following
    T(p) = p.A
    p = [c b a]
     
  4. Dec 26, 2011 #3

    Deveno

    User Avatar
    Science Advisor

    evidently, the "standard basis" being referred to is {1,t,t2}.

    that is: a+bt+ct2<---> (a,b,c)

    (1,0,0) <--> 1
    (0,1,0) <--> t
    (0,0,1) <--> t2

    with respect to a given basis, B = {v1,...,vn}, the matrix of a linear transformation T:V-->V, [T]B, is the matrix such that:

    [T]B[vj]B = [T(vj)]B,

    where the multiplication on the left is an nxn matrix multiplied by an nx1 matrix.

    of course, [vj]B = [0,0,...1,...,0]B, where the 1 is in the j-th place, which then gives us the j-th column of [T]B.
     
  5. Dec 28, 2011 #4
    Does this case of "standard basis" only apply for any 3-element vector input written in the form of a parabolic equation ?
     
  6. Dec 28, 2011 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Yes. Your "standard basis", <1, 0, 0>, <0, 1, 0>, and <0, 0, 1>, would be correct for [itex]R^3[/itex]. Of course, the assignment <1, 0, 0>->1, <0, 1, 0>->t, <0, 0, 1>->[itex]t^2[/itex] maps one to the other.
     
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