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Math Help - Is the linear transformation y = A(Bx) invertible?

  1. #1
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    Is the linear transformation y = A(Bx) invertible?

    I'm having trouble with this question.



    Here is what I have:

    From the hint, we solve the equation \vec{y}=A(B\vec{x}) first for B(\vec{x}) and then for \vec{x}. So we have

    B\vec{x}=A^{-1}\vec{y}

    so...

    \vec{x}=B^{-1}A^{-1}\vec{y}

    I'm really not sure where to go from here though.
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  2. #2
    A Plied Mathematician
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    Well, what does the fact that both A and B are invertible (so you can write their inverses) say about the last equation you wrote down? Is the transformation invertible?
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  3. #3
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    Ok, I think I understand.

    So it is invertible and its inverse is simply:

    \vec{x} = B^{-1}A^{-1}\vec{y}

    Is that correct?
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  4. #4
    A Plied Mathematician
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    I suppose your answer is in keeping with the question's nomenclature in the OP. I would say that AB is the linear transformation, and that

    (AB)^{-1}=B^{-1}A^{-1}

    is the inverse transformation.

    I think you get the main mathematical idea, though.
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