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A linear combination of vectors is a vector sum of these vectors rescaled (i.e. multiplied) with any scalars $(x_1,x_2,x_3)$. For example, taking scalars $(2,1,-3)=x$, we can find a vector $b$ for which the system $Ax=b$ would have a solution vector $x$.

$$\begin{aligned} b &= 2C_1+C_2-3C_3 \\ &=\begin{pmatrix} -2\\ 17\\ -8\end{pmatrix} \end{aligned}$$

We can check by direct substitution that $x = \{2, 1, -3\}^T$ does represent a solution vector.

$$\begin{aligned} Ax &= b\\ \begin{pmatrix} -2\\ 17\\ -8\end{pmatrix}&=\begin{pmatrix} -2\\ 17\\ -8\end{pmatrix} \end{aligned}$$

Sorry for formatting, I don't know how to make augmented matrices.

How does this make sense? What does it all mean? I know what augmented matrices are and vectors, but nothing beyond that. Why is the augmented matrix $\{-2,17,-8\}$? How did he get that answer?

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  • $\begingroup$ Welcome to Math.StackExchange. When you post questions/comments/answers, please use $\LaTeX$ formatting to make your questions more readable. Also, as is, you have not defined what the matrix $A$ is, so nobody can answer this question for you. Google "latex pmatrix" to figure out how to write up a matrix on this site. $\endgroup$ – NoseKnowsAll Sep 18 '15 at 15:41
  • $\begingroup$ I've edited your post to make it better, but as previously mentioned, without defining what $A$ is, nobody can answer this question. Feel free to comment back when you have edited this post with what $A$ is in latex's pmatrix format (as I have re-written your vectors already), and I'll answer your concerns. $\endgroup$ – NoseKnowsAll Sep 18 '15 at 15:48
  • $\begingroup$ To see how to typeset matrix, have a look here and here. $\endgroup$ – Martin Sleziak Sep 18 '15 at 19:11

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