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Math Help - Finding a formula for a linear map.

  1. #1
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    Angry Finding a formula for a linear map.

    I have a question here that states:

    Find a formula for the linear map T: P2(R) --> P1(R) such that

    M(T, v1, v2, v3, w1, w2) = (the matrix) 1 -2 4
    0 2 3

    Here v1(x) = 1 + x, v2(x) = x, v3(x) = x - 1, w1(x) = 1 - x and w2(x) = 4 + x

    I hope i made that clear enough, tried my best.

    This is what I have come up with:

    T(v1) = 1(w1) + 0(w2) = 1 - x
    T(v2) = -2(w1) + 2(w2)
    = (-2 + 2x) + (8 + 4x) = -2 + 8 +2x + 4x
    = 6 + 6x
    T(v3) = 4(w1) + 3(w2)
    = (4 - 4x) + (12 + 3x) = 4 - 4x + 12 + 3x
    = 16 - x

    T(x) = ?

    v1 = 1 + x ==> T(v1) = 1 - x
    v2 = X ==> T(v2) = 6 + 6x
    v3 = x + 1 ==> T(v3) = 16 - x

    T(x) ==> ?


    Could anyone help me from there? Or if I am completely wrong, could someone put me in the right direction? I appreciate any help. Thanks.
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  2. #2
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    I should also add that the 1, -2, 4; 0, 2, 3 is a 2x3 matrix. With the first set of numbers being in top and the second set on bottom. Sorry for any confusion.
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  3. #3
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    Joined
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    Just curious..

    Did you come up with a solution to this problem yet? I am curious as to what the answer is
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  4. #4
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    Quote Originally Posted by GreenDay14 View Post
    I have a question here that states:

    Find a formula for the linear map T: P2(R) --> P1(R) such that

    M(T, v1, v2, v3, w1, w2) = (the matrix) 1 -2 4
    0 2 3

    Here v1(x) = 1 + x, v2(x) = x, v3(x) = x - 1, w1(x) = 1 - x and w2(x) = 4 + x

    I hope i made that clear enough, tried my best.

    This is what I have come up with:

    T(v1) = 1(w1) + 0(w2) = 1 - x
    T(v2) = -2(w1) + 2(w2)
    = (-2 + 2x) + (8 + 4x) = -2 + 8 +2x + 4x
    = 6 + 6x
    T(v3) = 4(w1) + 3(w2)
    = (4 - 4x) + (12 + 3x) = 4 - 4x + 12 + 3x
    = 16 - x

    T(x) = ?

    v1 = 1 + x ==> T(v1) = 1 - x
    v2 = X ==> T(v2) = 6 + 6x
    v3 = x + 1 ==> T(v3) = 16 - x

    T(x) ==> ?


    Could anyone help me from there? Or if I am completely wrong, could someone put me in the right direction? I appreciate any help. Thanks.
    I trust you have completed it. What else is left? Once you know how to map the basis you know the linear transformation completely.

    Express x as linear combination of v1,v2,v3.
    Apply T.
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