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If I have a set of linearly independent equations, E1, E2, ..., En. I am wondering if all the possible sums are linearly independent between them and between the original equations.

{E1, E2, .., En, E1+E2, E1+E3, E2+E3, ..., Ei+En} are linearly independent?

If yes, how can I prove it? Thanks.

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Of course not (unless $n=1$). Since your set contains $E_1$, $E_2$ and $E_1+E_2$, it cannot be linearly independent, because$$1\times E_1+1\times E_2+(-1)\times(E_1+e_2)=0.$$

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  • $\begingroup$ Thank you, and what about this set of equations? {E1, E1+E2, E1+E3, E1+E4} are they linearly independent? How do I prove this? $\endgroup$ – gen Oct 6 '17 at 12:38
  • $\begingroup$ @gen Which set of equations? $\endgroup$ – José Carlos Santos Oct 6 '17 at 12:39
  • $\begingroup$ Sorry I edited my comment. Thanks. $\endgroup$ – gen Oct 6 '17 at 12:42
  • $\begingroup$ @gen That's another question. If you want to ask that question, ask it as a separate question. And if my answer was useful, perhaps that you could mark it as your accepted answer. $\endgroup$ – José Carlos Santos Oct 6 '17 at 12:44
  • $\begingroup$ Thank you. I really appreciate your help. I've asked a new question here: math.stackexchange.com/questions/2460334/… $\endgroup$ – gen Oct 6 '17 at 12:50

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