Exercise 2.A.10

Question

Suppose $v_1,v_2,v_3,v_4$ spans $V$. Prove that the list
$$v_1 - v_2, v_2 - v_3, v_3 - v_4, v_4$$
also spans $V$.

Célio Passos · Accepted Answer

\begin{proof}
Suppose that $v_1 + w, \dots, v_m + w$ is linearly dependent. There are $a_1, \dots, a_m \in \mathbb{F}$ not all zero such that

$$ \begin{aligned} a_1 (v_1 + w) + \dots + a_m (v_m + w) &= 0\ a_1 v_1 + \dots + a_m v_m &= - (a_1 + \dots + a_m) w\ -\frac{a_1}{a_1 + \dots + a_m} v_1 - \dots -\frac{a_m}{a_1 + \dots + a_m} v_m &= w\ \end{aligned} $$

Because $v_1, \dots, v_m$ is linearly independent, it follows by the second equation that $a_1 + \dots + a_m 
eq 0$. Hence the third equation makes sense and $w \in \operatorname{span}(v_1, \dots, v_m)$.
\end{proof}

Exercise 2.A.10

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Exercise 2.A.10

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