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Given the vectors $v_1 = (2,1,0,3)$, $v_2 = (3,-1,5,2)$ and $v_3=(-1,0,2,1)$, verify that the following vectors are in the span of ${v_1,v_2,v_3}$
a) $(0,0,0,0)$
b) $(1,1,1,1)$

For b) I reduced the following matrix:
$\left[ \begin{array}{ccc|c} 2 & 3 & -1 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & 5 & 2 & 1 \\ 3 & 2 & 1 & 1 \end{array} \right]$

and got

$\left[ \begin{array}{ccc|c} 1 & 4 & -1 & 0 \\ 0 & -5 & 1 & 1 \\ 0 & 0 & 3 & 2 \\ 0 & 0 & 2 & -1 \end{array} \right]$

Since the last two rowse are inconsistent, the system is inconsistent. So that means that $(1,1,1,1)$ is not in the span of ${v_1,v_2,v_3}$ doesn't it?

For a) I'm thinking I don't even have to do gaussian elimination because the trivial linear combination of these vectors will result in $(0,0,0,0)$ and it will be in the span of ${v_1,v_2,v_3}$. Is that right? Thanks.

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Your reasonings for a) and b) are correct.

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  • $\begingroup$ Ok, thanks @Fred. $\endgroup$
    – Bucephalus
    Aug 15 '17 at 12:42

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