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Math Help - Determining matrix relationship of linear transformations

  1. #1
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    Determining matrix relationship of linear transformations

    P' is the point of intersection of the line y=x and the line of slope 2 that passes through P.

    [Sol] Letting (a,b) and (x',y') be the coordinates of P and P', respectively,

    Calculating the equation of the line of slope 2 that passes through P.

    y-b=2(x-a)
    y=2x-2a+b

    Since the lines y=x and y=2x-2a+b intersect at P'(x',y'),

    x'=2a-b
    y'=2a-b (That's my problem, how do you get x' and y'?)
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  2. #2
    Super Member Gamma's Avatar
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    Well you are solving the system of equations
    y=2x-2a+b
    x=y

    so just substitute to get
    x=2x-2a+b \Rightarrow x=2a+b
    but then you have x=y, so this is also the y coordinate

    This gives you the system you have.
    Last edited by Gamma; August 19th 2009 at 07:00 PM. Reason: TeX error
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  3. #3
    Super Member Gamma's Avatar
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    I didn't read the title of this, if you want them to be solved as like a matrix equation, you need to first put the system in standard form.

    so you have
    y=2x-2a+b
    x=y

    This yields
    -2x+y=-2a+b
    x-y =0

    Which is the same as
    \begin{pmatrix}-2 & 1 \\ 1 &-1 \end{pmatrix} \begin{pmatrix}x\\y \end{pmatrix}=\begin{pmatrix}-2a+b\\0 \end{pmatrix}

    So then you can solve this by row reduction, or finding the inverse of this matrix, or whatever method you want, you will get the same solution as above.
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