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Can someone help me to solve this question please :

Establish, by induction, that : $ \forall n \in \mathbb{N} \setminus \{ 0,1 \} \ \forall v_1 , \dots , v_n \in \mathbb{R}^n $ linearly independants :

$ \mathcal{B}_{C_n^2} = \{ (v_1 - v_2) , (v_1 - v_3) , \ \dots \ , (v_1 - v_n) , ( v_2 - v_3 ) , ( v_2 - v_4 ) , \ \dots \ , (v_2 - v_n) , \ \dots \ , ( v_{n-1} - v_n ) \} $

is a family of linearly dependent vectors of $ \mathbb{R}^n $.

Thanks in advance for your help.

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  • $\begingroup$ Well, actually in $\Bbb R^n$ any more than $n$ vectors are linearly dependent. $\endgroup$ – Berci Jul 20 '16 at 21:42
  • $\begingroup$ Thank you. It's easy as a question. :) $\endgroup$ – Lina45 Jul 20 '16 at 21:45
  • $\begingroup$ The statement is wrong for $n = 2$. $\endgroup$ – Jendrik Stelzner Jul 20 '16 at 21:48
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Just pick a telescoping sum, then add in the correction term: $(v_1-v_2)+(v_2-v_3)+\cdots+(v_{n-1}-v_n)-(v_1-v_n)=0$

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  • $\begingroup$ Yes. Thank you very much. :) $\endgroup$ – Lina45 Jul 20 '16 at 21:53

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