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Math Help - building a transformation from the image

  1. #1
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    building a transformation from the image

    give an example for a transformation T:R^3 -> R^3
    in which (ImT)^{\perp}=\{(x,y,z)|x-3y+2z=0

    and find T(x,y,z) formula

    ??

    i solvled this x-3y+2z=0 equation
    i got two vectors then i took V=(a,b,c)
    and said that V*U1=0 and V*U2=0
    and that v is ImT

    the result is Im T=sp{(-1,3,0)}

    i got 2 thing to do here
    the first one:
    T(0,0,1)=(-1,3,0)
    T(0,1,0)=(-1,3,0)
    T(1,0,0)=(-1,3,0)
    correct?

    regarding the second one:
    how to find T(x,y,z) formula
    ?
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  2. #2
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    re: building a transformation from the image

    (-1,3,0) cannot be in the image of T. Consider the element (1,1,1)\in (\mathrm{im} T)^{\perp}, for example.

    (-1,3,0)\cdot (1,1,1)=2\neq 0\Rightarrow (-1,3,0)\notin \mathrm{im} T.

    The correct answer, I believe, is that \mathrm{im} T=\mathrm{span}((1,-3,2)). I got this just by writing out a few elements of (\mathrm{im} T)^{\perp}, noting that if an element is in the image of T then its dot product with each of those elements must be 0, and solving a system of equations based on this fact.

    As far as finding a formula for T, I'm not 100% sure what to do. I mean, we can define a lot of different linear transformations whose image is \mathrm{span}((1,-3,2)). I guess just pick one.
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  3. #3
    Senior Member Tinyboss's Avatar
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    re: building a transformation from the image

    The set \{(x,y,z)|x-3y+2z=0\} is just the set of vectors (x,y,z) which have zero dot product with the vector (1,-3,2), in other words, the orthogonal complement of <1,-3,2>. You want a linear transformation with that set as its orthogonal complement, that is, you need the image to be <1,-3,2>. Just send one or more basis elements to nonzero scalar multiples of (1,-3,2), and send the rest to zero.
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  4. #4
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    re: building a transformation from the image

    ahhhh
    t(0,0,1)=(1,-3,2)
    t(0,1,0)=(0,0,0)
    t(1,0,0)=(0,0,0)
    correct?
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  5. #5
    Senior Member Tinyboss's Avatar
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    re: building a transformation from the image

    Yes, that's one correct choice.
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