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Math Help - Using Matrices to Find Basis of Image of Linear Transformation

  1. #1
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    Using Matrices to Find Basis of Image of Linear Transformation

    Hi all ---

    I'm having trouble understanding one part of this example from my textbook. It's using the matrix of a linear transformation to find the basis of the image of the same linear transformation.

    Question ---

    Let L:{{P}_{2}}\to {{M}_{2\times 2}} be a linear transformation defined by:

     L\left( {{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}} \right)=\left[ \begin{matrix} {{a}_{2}} & -{{a}_{2}} \\ {{a}_{0}}-{{a}_{2}} & {{a}_{0}}+{{a}_{2}} \end{matrix} \right] .

    Let A = {x^2 - x, x - 1, x^2 + 1} be a basis for P_2 and
    let B = {$\left[ \begin{matrix}<br />
1 & 0 \\ 0 & 1 \end{matrix} \right],\left[ \begin{matrix}<br />
0 & 1 \\ 1 & 0 \end{matrix} \right],\left[ \begin{matrix}<br />
0 & 0 \\ 1 & 1 \end{matrix} \right],\left[ \begin{matrix}<br />
1 & 1 \\ 0 & 0 \end{matrix} \right]$<br />
} be a basis for M_{2 \times 2}.

    If ${{\left[ T \right]}_{BA}}=\left[ \begin{matrix}<br />
1 & 0 & 1 \\ -1 & 0 & -1 \\ 0 & -1 & 1 \\ 0 & 0 & 0 \end{matrix} \right]$, use it to find the basis for the image of L.

    Textbook Solution ---

    Recall that v is in im L \Leftrightarrow {{\left[ w \right]}_{B}}\in col\left( {{\left[ T \right]}_{BA}} \right) .

    But [T]_{BA} is already in row-reduced format,
    so \col\left( {{\left[ T \right]}_{BA}} \right)=span\left\{ \left[ \begin{matrix}<br />
1 \\ -1 \\ 0 \\ 0 \end{matrix} \right],\left[ \begin{matrix}<br />
0 \\ 0 \\ -1 \\ 0 \end{matrix} \right] \right\}\<br />

    This is what I don't understand. How do you know that the vectors in the span here are coordinate vectors? An obvious answer - but not the one I'm looking for - is because these vectors aren't in P_2 or M_{2 \times 2}. But what's the true math answer? Thanks a lot ---

    so   \[\text{image }T=span\left\{ \left( \left[ \begin{matrix}<br />
   1 & 0  \\   0 & 1  \end{matrix} \right],\left[ \begin{matrix}<br />
   0 & -1  \\   -1 & 0  \end{matrix} \right] \right),\left[ \begin{matrix}<br />
   0 & 0  \\   -1 & -1  \end{matrix} \right] \right\}\]<br /> <br />
.
    Last edited by mathminor827; September 30th 2011 at 01:10 PM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Using Matrices to Find Basis of Image of Linear Transformation

    Hint: The linear transformation can be written in the way:



    where the column vectors span \textrm{Im}f and are expressed in coordinates on B .
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