\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
In \(\mathbb{R}^3\), set \(F = \{(x, y, z) \in \mathbb{R}^3 : x + y - z = 0\}\) and \(G = \{(x, y, z) \in \mathbb{R}^3 : 2 x - y + z = 0\}\). Show that \(F \cap G\) is a subspace of \(\mathbb{R}^3\) and exhibit a generating family.
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