# linearly dependent family of vectors.

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 $\mathbb{R}^n$.

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