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Math Help - Solution Space

  1. #1
    Junior Member
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    Solution Space

    I am currently learning about linear transforms and came across this problem:

    Given a matrix: A =
    [ 4 3 -1 7 ]
    [ 1 1 0 2 ]
    [ 2 5 3 7 ]

    We are told that to find all the values t such that b = [ 1 t t^2 ]^T is solvable.

    My attempt: I augmented the matrix to,

    [ 4 3 -1 7 | 1 ]
    [ 1 1 0 2 | t ]
    [ 2 5 3 7 | t^2]

    and ended up with:

    [ 1 0 0 0 | t^2 -6t +1 ]
    [ 0 1 2 0 | t^2 + t - 1 ]
    [ 0 0 1 -1| t^2 - 3t ]

    I'm not sure how to proceed from here. I understand that x_1 = t^2 - 6t + 1, that x_2 + 2 x_3 = t^2 + t - 1, and that x_3 - x_4 = t^2 - 3t. But how do those equations affect the constraints on t? It seems that no matter what I plug in for my free variable x_4, I can plug in whatever I want for t.
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
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    Yuma, AZ, USA
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    Re: Solution Space

    Quote Originally Posted by jsndacruz View Post
    I am currently learning about linear transforms and came across this problem:

    Given a matrix: A =
    [ 4 3 -1 7 ]
    [ 1 1 0 2 ]
    [ 2 5 3 7 ]

    We are told that to find all the values t such that b = [ 1 t t^2 ]^T is solvable.

    My attempt: I augmented the matrix to,

    [ 4 3 -1 7 | 1 ]
    [ 1 1 0 2 | t ]
    [ 2 5 3 7 | t^2]

    and ended up with:

    [ 1 0 0 0 | t^2 -6t +1 ]
    [ 0 1 2 0 | t^2 + t - 1 ]
    [ 0 0 1 -1| t^2 - 3t ]

    I'm not sure how to proceed from here. I understand that x_1 = t^2 - 6t + 1, that x_2 + 2 x_3 = t^2 + t - 1, and that x_3 - x_4 = t^2 - 3t. But how do those equations affect the constraints on t? It seems that no matter what I plug in for my free variable x_4, I can plug in whatever I want for t.
    You need to double check your work.

    After a few row operations I get the matrix

    \begin{bmatrix}1 & 1 & 0 & 2 & t \\ 0 & -1 & -1 & -1 & 1-4t \\ 0 & 0 & 0 & 0 & t^2-14t+3 \end{bmatrix}

    To have a solution we must have t^2-14t+3=0
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  3. #3
    Junior Member
    Joined
    Jun 2011
    Posts
    45

    Re: Solution Space

    Aha - silly mistake. I always seem to make them when doing reef. Thanks so much for your help!
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