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Math Help - More linear Algebra

  1. #1
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    More linear Algebra

    Find an example of a linear transformation Transpose: R^4--->R^3 such that
    ker T= span{(1,0,0,0),(0,0,1,0)}
    and
    im T= span{(1,0,0),(0,0,1)}.

    Fully justify example.
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  2. #2
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    Quote Originally Posted by natewalker205 View Post
    Find an example of a linear transformation Transpose: R^4--->R^3 such that
    ker T= span{(1,0,0,0),(0,0,1,0)}
    and
    im T= span{(1,0,0),(0,0,1)}.

    Fully justify example.
    The transformation is uniquely defined by the images of the unit vectors in R^4, which are (1,0,0,0)^t,\ (0,1,0,0)^t,\ (0,0,1,0)^t,\ (0,0,0,1)^t

    Now:

    <br />
{\rm{ker}}(T)={\rm{span}} \{(1,0,0,0)^t,\ (0,0,1,0)^t \}<br />

    tells us that the images of these two vectors are both (0,0,0)^t

    and as:

    <br />
{\rm{image}}(T)={\rm{span}} \{(1,0,0)^t,\ (0,0,1)^t \}<br />

    we can let these two vectors be the images of (0,1,0,0)^t and (0,0,0,1)^t respectivly.

    So now we have assigned images to the unit vectors in R^4 which satisfy all the given conditions.

    Then clearly we can write:

    <br />
T = \left[ {\begin{array}{*{20}c}<br />
   0 & 1 & 0 & 0  \\<br />
   0 & 0 & 0 & 0  \\<br />
   0 & 0 & 0 & 1  \\<br />
\end{array}} \right]<br />

    as one such matrix.
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