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

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

Attempt:

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

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

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

inconsistent.