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Math Help - Vectors

  1. #1
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    Vectors

    Given \bold{v_1} = \left[ \begin {array}{c} 1\\\noalign{\medskip}1\\\noalign{\medskip}1\end {array} \right], \bold{v_2} = \left[ \begin {array}{c} 1\\\noalign{\medskip}0\\\noalign{\medskip}-1\end {array} \right], \bold{v_3} = \left[ \begin {array}{c} 2\\\noalign{\medskip}1\\\noalign{\medskip}3\end {array} \right]

    Determine a vector \bold{w} in \mathbb{R}^3 (which are NOT scalars of \bold{v_1,v_2, v_3} thats in the Span{ \bold{v_1},\bold{v_2},\bold{v_3}} but is NOT in Span{ \bold{v_1},\bold{v_2}}. Justify your answer.

    Attempt:

    So I think what they want me to do is find values a, b, c for:

    \left[ \begin {array}{cccc} 1&1&2&a\\\noalign{\medskip}1&0&1&b\\\noalign{\meds  kip}1&-1&3&c\end {array} \right] that make it consistent but what make

     \left[ \begin {array}{ccc} 1&1&a\\\noalign{\medskip}1&0&b\\\noalign{\medskip}  1&-1&c\end {array} \right]

    inconsistent.
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  2. #2
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    Anyone know of such vector? I've played around with it and I can get a vector that works for the first but it also works for the second.
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by LinAlg View Post
    Given \bold{v_1} = \left[ \begin {array}{c} 1\\\noalign{\medskip}1\\\noalign{\medskip}1\end {array} \right], \bold{v_2} = \left[ \begin {array}{c} 1\\\noalign{\medskip}0\\\noalign{\medskip}-1\end {array} \right], \bold{v_3} = \left[ \begin {array}{c} 2\\\noalign{\medskip}1\\\noalign{\medskip}3\end {array} \right]

    Determine a vector \bold{w} in \mathbb{R}^3 (which are NOT scalars of \bold{v_1,v_2, v_3} thats in the Span{ \bold{v_1},\bold{v_2},\bold{v_3}} but is NOT in Span{ \bold{v_1},\bold{v_2}}. Justify your answer.

    Attempt:

    So I think what they want me to do is find values a, b, c for:

    \left[ \begin {array}{cccc} 1&1&2&a\\\noalign{\medskip}1&0&1&b\\\noalign{\meds  kip}1&-1&3&c\end {array} \right] that make it consistent but what make

     \left[ \begin {array}{ccc} 1&1&a\\\noalign{\medskip}1&0&b\\\noalign{\medskip}  1&-1&c\end {array} \right]

    inconsistent.
    ok, you did well setting up the matrices.

    note that the solution to the first is:

    \left[ \begin{array}{cccl} 1 & 0 & 0 & -a + 3b - c \\ 0 & 1 & 0 & b - c \\ 0 & 0 & 1 & a - 2b + c \end{array} \right]

    (you should probably check this, i always mess up the arithmetic somewhere because i try to do it in my head )

    the solution to the second is:

    \left[ \begin{array}{ccl} 1 & 0 & b \\ 0 & 1 & a - b \\ 0 & 0 & {\color{red}a - 2b + c} \end{array} \right]

    hmm, something looks familiar here.

    note that the latter matrix is inconsistent if a - 2b + c \ne 0. so choose a,b, and c that work for the first solution but make a - 2b + c \ne 0 and you can find your vector.
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