# Thread: another linear dependence problem

1. ## another linear dependence problem

Let the set {v1, v2, v3} be linearly dependent. Prove that {v1, v1+v2, V3} is also linearly dependent.

Your help would be greatly appreciated. thank you!

2. the Given two vector set are equavalent, thus if the second set is linear independent, then the first set is also linear independent.

3. following up Shanks words, let's use $\displaystyle p\implies q\equiv\sim q\implies\sim p.$

$\displaystyle av_1+bv_2+cv_3=(a+b)v_1+bv_2+cv_3-bv_1,$ but $\displaystyle \{v_1,v_1+v_2,v_3\}$ is a linear independent set, thus $\displaystyle a+b=b=c=0,$ and $\displaystyle a=b=c=0,$ so $\displaystyle \{v_1,v_2,v_3\}$ is a linear independent set.

4. Originally Posted by Krizalid
following up Shanks words, let's use $\displaystyle p\implies q\equiv\sim q\implies\sim p.$

$\displaystyle av_1+bv_2+cv_3=(a+b)v_1+bv_2+cv_3-bv_1,$ but $\displaystyle \{v_1,v_1+v_2,v_3\}$ is a linear independent set, thus $\displaystyle a+b=b=c=0,$ and $\displaystyle a=b=c=0,$ so $\displaystyle \{v_1,v_2,v_3\}$ is a linear independent set.
i think he's trying to prove that the system $\displaystyle \left \{ v_{1},v_{1}+v_{2},v_{3} \right \}$ is linearly dependent.
non ?

5. no, he's trying to prove that the second set is linearly dependent.

6. the system $\displaystyle \left \{ v_1,v_1+v_2,v_3 \right \}$ is linearly dependent if there's some scalars$\displaystyle \lambda_1,\lambda _2$ and $\displaystyle \lambda_3$ not all zeros such that,
$\fn_cs \lambda _1v_1+\lambda _2(v_1+v_2)+\lambda_3v_3=0$.
which give us, $\fn_cs \lambda _1v_1+\lambda _2v_1+\lambda_2v_2+\lambda_3v_3=0$
$\fn_cs \Rightarrow (\lambda _1+\lambda _2)v_1+\lambda_2v_2+\lambda_3v_3=0$
setting $\fn_cs \lambda_1 +\lambda_2 =\lambda '_1$
$\fn_cs \Rightarrow \lambda' _1v_1+\lambda_2v_2+\lambda_3v_3=0$
and since the system $\fn_cs \left \{ v_1,v_2,v_3 \right \}$ is linearly dependent,the scalars aren't all zeros,therefore the system $\displaystyle \left \{ v_1,v_1+v_2,v_3 \right \}$ is linearly dependent.