1. ## Vectors

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.

2. 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.

3. Originally Posted by 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.
ok, you did well setting up the matrices.

note that the solution to the first is:

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

(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:

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

hmm, something looks familiar here.

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