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Math Help - Interesting Linear Transformation question

  1. #1
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    Interesting Linear Transformation question

    *** U \text{  and  } W \in R_4[x]
    and below I simply transform from  R_4[x] \rightarrow R^4

    W = (i,j,0,-j) \phantom{hacklol} U = (4s, 4t, -s, -t)

    Does there exist a linear transformation T:R_4[x] \rightarrow R_4[x] so that T(U)=W and T(W) = U

    So I got to this...

    W = Sp{(1,0,0,0),(0,1,0,-1)}
    U = Sp{(4,0,-1,0),(0,4,0,-1)}

    And if I prove it for the bases, I prove it for the entire thing. But how do I continue from here?

    I know in general that if you have a basis you can create any transformation you want but I don't know how to.

    Thanks!
    Last edited by jayshizwiz; November 3rd 2010 at 01:54 AM.
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  2. #2
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    Ok, I think I got it. It took me many pages on my notebook...

    Basically, T(x,y,z,r) = (4x+15z, -5r-y, -x-4z, r)

    Thanks anyway!

    ***and after that I must change it back to R_4[x]...
    Last edited by jayshizwiz; November 3rd 2010 at 01:57 AM.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by jayshizwiz View Post
    W = (i,j,0,-j) \phantom{hacklol} U = (4s, 4t, -s, -t)

    Does there exist a linear transformation T:R_4[x] \rightarrow R_4[x] so that T(U)=W and T(W) = U

    So I got to this...

    W = Sp{(1,0,0,0),(0,1,0,-1)}
    U = Sp{(4,0,-1,0),(0,4,0,-1)}

    And if I prove it for the bases, I prove it for the entire thing. But how do I continue from here?

    I know in general that if you have a basis you can create any transformation you want but I don't know how to.

    Thanks!
    What exactly are these spaces?
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  4. #4
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    Oy! Sorry, I always do that. I try to save time so I just give half the question... Maybe later, if I have more time, I'll just post the entire question with my answer and hopefully itwill be right.

    I edited it at the top of my original post. Both spaces are in R_4[x]. Sorry for the confusion
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by jayshizwiz View Post
    Oy! Sorry, I always do that. I try to save time so I just give half the question... Maybe later, if I have more time, I'll just post the entire question with my answer and hopefully itwill be right.

    I edited it at the top of my original post. Both spaces are in R_4[x]. Sorry for the confusion
    Haha, the funny this is my problem was not knowing what R_4[x] is? Maybe \mathbb{R}_4[x]=\left\{a_0+\cdots+a_4x^4:a_0,\cdots,a_4\in\mathbb  {R}\right\} but from the elements I would assume not.
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  6. #6
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    R_4[x] = a_0 + a_1x+a_2x^2+a_3x^3 : a_0,...,a_3 \in R
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