\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Let \(E\) be a \(\mathbb{K}\)-vector space and \(u_1, \ldots, u_n \in E\). For \(k \in \llbracket 1, n \rrbracket\), set \(v_k = u_1 + u_2 + \cdots + u_k\).
  1. Show that the family \((u_1, \ldots, u_n)\) is free if and only if the family \((v_1, \ldots, v_n)\) is free.
  2. Show that \((u_1, \ldots, u_n)\) generates \(E\) if and only if \((v_1, \ldots, v_n)\) generates \(E\).

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