Let the set {v1, v2, v3} be linearly dependent. Prove that {v1, v1+v2, V3} is also linearly dependent.
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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.
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,
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which give us,
setting
and since the system 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.