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Thread: Linear combination problem

  1. #1
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    Linear combination problem

    "Show that

    [1]
    [1]
    [1] can't be written as a linear combination of

    [1]
    [0]
    [1]
    and
    [0]
    [1]
    [1] "


    I'm not sure where to even start here. Sorry about the formatting I don't know how to make matrices.
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  2. #2
    MHF Contributor

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    Re: Linear combination problem

    Quote Originally Posted by MrJank View Post
    "Show that

    [1]
    [1]
    [1] can't be written as a linear combination of

    [1]
    [0]
    [1]
    and
    [0]
    [1]
    [1] "


    I'm not sure where to even start here. Sorry about the formatting I don't know how to make matrices.
    Suppose that $\alpha \left[ {\begin{array}{*{20}{c}}
    1 \\
    0 \\
    1
    \end{array}} \right] + \beta \left[ {\begin{array}{*{20}{c}}
    0 \\
    1 \\
    1
    \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
    1 \\
    1 \\
    1
    \end{array}} \right]$ so that $\alpha=1,~\beta=1~\&~\alpha+\beta=1$

    Now what?
    Thanks from topsquark
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  3. #3
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    Re: Linear combination problem

    Well, if a = 1 and b = 1, a + b = 1 is a false statement.

    Is that the answer to the question? I didn't think it would be easy as that...
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  4. #4
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    Re: Linear combination problem

    Quote Originally Posted by MrJank View Post
    Well, if a = 1 and b = 1, a + b = 1 is a false statement.
    Is that the answer to the question? I didn't think it would be easy as that...
    Well we supposed that $\left[ {\begin{array}{*{20}{c}}
    1 \\
    1 \\
    1
    \end{array}} \right] = \alpha \left[ {\begin{array}{*{20}{c}}
    1 \\
    0 \\
    1
    \end{array}} \right] + \beta \left[ {\begin{array}{*{20}{c}}
    0 \\
    1 \\
    1
    \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
    \alpha \\
    \beta \\
    {\alpha + \beta }
    \end{array}} \right] \Rightarrow \begin{array}{*{20}{r}}
    {\alpha = 1} \\
    {\beta = 1} \\
    {\alpha + \beta = 1}
    \end{array}$ That is a condiction.
    So it is proof by contradiction,
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