\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Let \(E\) be a \(\mathbb{K}\)-vector space and \((u_1, \ldots, u_n)\) a free family of \(E\). Let \(u_{n+1} \in E\). Show that \((u_1, \ldots, u_n, u_{n+1})\) is free if and only if \(u_{n+1} \notin \mathrm{Vect}(u_1, \ldots, u_n)\).
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