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Math Help - Linear Operators

  1. #1
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    Linear Operators

    Show that the range of the linear operator defined by the equations
    w1 = 4x1 - 2x2
    w2 = 2x1 - x2

    is not all of R2, and find a vector that is not in the range.


    Ok I can easily show the range is not all of R2 by finding the determinant to be zero... but how the vector which is not in the range? I know this is probably really easy.
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi

    <br />
w_1 =4x_1-2x_2 and <br />
w_2  =2x_1-x_2<br />
\implies <br />
w_1=2w_2<br />
\implies\begin{pmatrix}w_1\\w_2 \end{pmatrix}= w_2\begin{pmatrix}2\\1 \end{pmatrix}

    All the vectors of the range can be written as \lambda \begin{pmatrix}2\\1 \end{pmatrix} hence, if you manage to find a vector of \mathbb{R}^2 which direction is not that of  \begin{pmatrix}2\\1 \end{pmatrix}, you are sure it does not belong to the range of the operator.
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  3. #3
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    Okay I get it, thanks so much
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