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Math Help - 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:

    T:V_3 \to V_3 such that T(0,1,2) = (3,1,2) and 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  T is a linear transformation, then there exists a transformation matrix  A for the mapping such that A\vec{x} = \vec{y}. It's clear that such a matrix must be 3x3.

    Let 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 \vec{x} with A and equate them to \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?
    <br />
\left[<br />
\begin {array}{ccc}<br />
0&1&1\\<br />
\noalign{\medskip}<br />
1&1&0\\<br />
\noalign{\medskip}<br />
1&0&1<br />
\end {array}<br />
\right]<br />
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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?
    <br />
\left[<br />
\begin {array}{ccc}<br />
a&1-2a&1+a\\<br />
\noalign{\medskip}<br />
1&1&0\\<br />
\noalign{\medskip}<br />
1&0&1<br />
\end {array}<br />
\right]<br />
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