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The homework question I have is:

Here are some vectors.

$\begin{bmatrix} 1\\1\\-2\end{bmatrix},\begin{bmatrix} 1\\2\\-2\end{bmatrix},\begin{bmatrix} 2\\7\\-4\end{bmatrix},\begin{bmatrix} 5\\7\\-10\end{bmatrix},\begin{bmatrix} 12\\17\\-24\end{bmatrix}$

Describe the span of these vectors as the span of as few vectors as possible.

Things I have tried:

  1. I looked in my book and on the web for similar examples. I've been studying the question at Find the span of a set of vectors quite a bit.In my problem, since all of these vectors have an $x,y,z$ component, any linear combination of the vectors will be in the form $\begin{bmatrix} x\\y\\z\end{bmatrix}\in\Bbb R^3$. So the $\text{span}$ of the 5 vectors above is equal to $\text{span}\{ \begin{bmatrix} 1\\1\\1\end{bmatrix}\}$?
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    $\begingroup$ No. The difference of the first 2 vectors in not in your proposed span. Hint: look carefully at the 1st and 3d entries in your 5 vectors. $\endgroup$ – kimchi lover Oct 22 '17 at 14:10
  • $\begingroup$ The third entry of each of the vectors is always multiplied by $-2$? I'm not sure i understand. $\endgroup$ – LovesPeanutButter Oct 23 '17 at 1:37
  • $\begingroup$ Is this true of the span of $(1,1,1)^T$? $\endgroup$ – kimchi lover Oct 23 '17 at 2:36

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