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Linear Transformation image

  1. Nov 27, 2009 #1
    1. The problem statement, all variables and given/known data

    Let v1=
    1
    -2
    and v2=
    -1
    1


    Let T:R2R2 be the linear transformation satisfying
    T(v1)=
    9
    7
    and T(v2)=
    0
    -8


    Find the image of an arbitrary vector
    x
    y



    2. Relevant equations



    3. The attempt at a solution

    I thought it might have to do something with T(u+v)=T(u)+T(v) or some sort of transformation, but I cannot seem to get it...
    Any help would be appreciated!
    Thanks!!
     
  2. jcsd
  3. Nov 28, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You are given that
    [tex]T\left(\begin{bmatrix}1 \\ -2\end{bmatrix}\right)= \begin{bmatrix} 9 \\ 7\end{bmatrix}[/tex]
    and that
    [tex]T\left(\begin{bmatrix}-1 \\ 1\end{bmatrix}\right)= \begin{bmatrix}0 \\ 8\end{bmatrix}[/tex]

    And you want to determine
    [tex]T\left(\begin{bmatrix} x \\ y\end{bmatrix}\right[/tex]

    Yes, you want to use T(u+v)= T(u)+ T(v). Specifically if [itex]u= Av_1+ Bv_2[/itex] then T(u)= AT(v_1)+ BT(v_2). So first you want find A and B such that
    [tex]\begin{bmatrix}x \\ y \end{bmatrix}= A\begin{bmatrix}1 \\-2\end{bmatrix}+ B\begin{bmatrix}-1 \\ 1 \end{bmatrix}[/tex]
     
  4. Nov 28, 2009 #3
    Alright, so I got
    A=-x-y
    B=-2x-y
    I'm guessing then we follow through with T(u)= AT(v_1)+ BT(v_2),

    T(x y)=[T(1 -2)T(0 -8)][A B]=[9A, 7A-8B]

    Then I sub in A and B:

    [9(-x-y), 7(-x-y)-8(-2x-y)]= [-9x-9y, 9x+y]

    Is this what I was supposed to do? I think now I have to factor out the x-y, but I can't do it to 9x+y. Did I do something wrong at finding A and B?
     
  5. Nov 29, 2009 #4
    I just submitted my work, it was right after all!!
    Thanks HallsofIvy!
     
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