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Math Help - Find Linear Transformations U and T such that...

  1. #1
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    Find Linear Transformations U and T such that...

    Attached is the problem statement and my attempt. I'm pretty sure my answers for the compositions are correct, but then again, I'm not sure why I can't find the correct matrices A and B from them (it looks like AB=BA=0 matrix, which contradicts the condition given in the problem statement).


    Can someone clarify this for me? ( Deveno? )

    Thanks!
    Attached Thumbnails Attached Thumbnails Find Linear Transformations U and T such that...-la3.jpg   Find Linear Transformations U and T such that...-la4.jpg  
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  2. #2
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    Re: Find Linear Transformations U and T such that...

    you do realize that T(a1,a2) = (a1+1,a2) is not a linear transformation, don't you?

    clearly T,U must be singular. an obvious singular transformation is T(x,y) = (x,x).

    can you find a U now so that UT = 0, but TU ≠ 0?
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  3. #3
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    Re: Find Linear Transformations U and T such that...

    you do realize that T(a1,a2) = (a1+1,a2) is not a linear transformation, don't you?
    I do now


    clearly T,U must be singular. an obvious singular transformation is T(x,y) = (x,x).
    I know singular means it doesn't have an inverse, but is there an easy way to tell right away if a transformation is singular or not?


    can you find a U now so that UT = 0, but TU ≠ 0?
    Well, it seems like U has to be something that outputs only zeros...so I gave U(x,y)=(0,0) a shot, but that messes up TU, because then TU=(0,0), which is what we don't want.


    Sooo...I switched things around and got T=(a2,0) and U=(0,a2)

    This seemed to work out nicely, although I'm curious if there was a way to do it with the transform you mentioned (ie, T(x,y)=(x,x)), because I couldn't get it to work with that one. What do you think?

    ps. thanks for the help!
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  4. #4
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    Re: Find Linear Transformations U and T such that...

    ways to tell if a transformation is singular:

    it's matrix (w.r.t. some bases) is not invertible (not so easy to tell).
    it's matrix (w.r.t. some bases) has a row of 0's at the bottom when row-reduced (this is a good general method).
    it's matrix (w.r.t. some bases) has 0 determinant (good for 2x2 and 3x3 matrices....bigger ones are harder).
    it's matrix (w.r.t. some bases) has more rows than columns.

    if it is possible to find U ≠ 0 with TU = 0.

    as for my example....there may have been other that would have worked, but i was thinking of U(x,y) = (x-y,0).

    then UT(x,y) = U(x,x) = (x-x,0) = 0, but:

    TU(x,y) = T(x-y,0) = (x-y,x-y) which is non-zero if x ≠ y.
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