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Math Help - proving if such transformation exists

  1. #1
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    proving if such transformation exists

    is there a transformation which follows these rules:
    T: R^4-> R^4
    T(-1,1,0,1)=(1,-1,2,1)
    T(1,1,1,0)=(2,3,1,-1)
    T(-1,5,2,3)=(3,7,0,-3)

    ImT is defined as span of the base of the images
    so i found that this group is dependant
    {(1,-1,2,1),(2,3,1,-1),(3,7,0,-3)}

    so ImT=Sp{(1,-1,2,1),(2,3,1,-1)}
    so dim ImT=2 and dim kerT=2

    so i took R^4 basis
    and defined T as:
    T(-1,1,0,1)=(1,-1,2,1)
    T(0,2,1,1)=(0,5,-3,-3)
    T(0,0,1,0)=(0,0,0,0)
    T(0,0,0,1)=(0,0,0,0)

    is it ok?
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  2. #2
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    Re: proving if such transformation exists

    Quote Originally Posted by transgalactic View Post
    is there a transformation which follows these rules:
    T: R^4-> R^4
    T(-1,1,0,1)=(1,-1,2,1)
    T(1,1,1,0)=(2,3,1,-1)
    T(-1,5,2,3)=(3,7,0,-3)
    You mean "linear transformation"- you make a general transformation that does anything!

    ImT is defined as span of the base of the images
    so i found that this group is dependant
    You mean "set". In "mathematical" English, "group" has a very specific meaning that is not correct here.

    {(1,-1,2,1),(2,3,1,-1),(3,7,0,-3)}
    Note that {(-1, 1, 0, 1), (1, 1, 1, 0), (-1, 5, 2, 3)} are also dependent. We can write
    -3(-1, 1, 0, 1)+ 2(1, 1, 1, 0)= (-1, 5, 2, 3).
    Is T(-1, 5, 2, 3)= -3T(-1, 1, 0, 1)+ 2T(1, 1, 1, 0)?

    so ImT=Sp{(1,-1,2,1),(2,3,1,-1)}
    so dim ImT=2 and dim kerT=2

    so i took R^4 basis
    I don't understand what you mean by this. What did you take as the basis? Are you referring to the standard basis?

    [tex]and defined T as:
    T(-1,1,0,1)=(1,-1,2,1)
    T(0,2,1,1)=(0,5,-3,-3)
    T(0,0,1,0)=(0,0,0,0)
    T(0,0,0,1)=(0,0,0,0)

    is it ok?
    Well, you haven't answered the question! What is your answer?
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  3. #3
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    Re: proving if such transformation exists

    a transformation of a basis is single.
    so if we write the transformation of a each vector in the R^4 basis
    then it defines a single transformation
    T(x,y,z,t)=xT(1,0,0,0)+yT(0,1,0,0)+zT(0,0,1,0)+tT( 0,0,0,1)

    ok there is no problem for me to find the value of
    T(0,0,0,1) etc..

    i ask if it ok to constract a transformation in such a way?

    how to know if it follows tle laws given in the question

    how do you see a definition of transformation?
    Last edited by transgalactic; June 24th 2011 at 11:09 AM.
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