\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
In \(\mathbb{R}^4\), set \(\mathbf{e}_1 = (1, 1, 1, 1)\), \(\mathbf{e}_2 = (1, 1, 0, 0)\), \(\mathbf{e}_3 = (1, 0, 0, 0)\), \(\mathbf{e}_4 = (0, 0, 0, 1)\). Show that \(\mathcal{B} = (\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3, \mathbf{e}_4)\) is a basis and determine the coordinates of \(v = (a, b, c, d)\) in \(\mathcal{B}\).
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