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Thread: Linear Mapping..

  1. #1
    Member kjchauhan's Avatar
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    Linear Mapping..

    Determine whether there exist a linear map in the following case:

    $\displaystyle T:V_3 \to V_3 $ such that $\displaystyle T(0,1,2) = (3,1,2)$ and $\displaystyle T(1,1,1)=(2,2,2)$

    If it exists give the general formula..

    Pl help to solve this problem..
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  2. #2
    Member Haven's Avatar
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    If $\displaystyle T $ is a linear transformation, then there exists a transformation matrix $\displaystyle A $ for the mapping such that $\displaystyle A\vec{x} = \vec{y}$. It's clear that such a matrix must be 3x3.

    Let $\displaystyle A = \left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{ 22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right]$

    And use the equations you have. Multiply the $\displaystyle \vec{x}$ with A and equate them to $\displaystyle \vec{y}$. You should have infinite solutions to the problem
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  3. #3
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    for existence...

    For existence, will this work?
    $\displaystyle
    \left[
    \begin {array}{ccc}
    0&1&1\\
    \noalign{\medskip}
    1&1&0\\
    \noalign{\medskip}
    1&0&1
    \end {array}
    \right]
    $
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  4. #4
    Member Haven's Avatar
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    Yes, that works fine. But the general forumla will be in terms of the a's of the matrix A
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  5. #5
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    a little more general

    Let's generalize the example a little. Will this work?
    $\displaystyle
    \left[
    \begin {array}{ccc}
    a&1-2a&1+a\\
    \noalign{\medskip}
    1&1&0\\
    \noalign{\medskip}
    1&0&1
    \end {array}
    \right]
    $
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