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Let \(A \in M_{n, p}(\mathbb{R})\). Show that the set \(\mathcal{S} = \{X \in \mathbb{R}^p : A X = 0\}\) of solutions of the homogeneous linear system \(A X = 0\) is a subspace of \(\mathbb{R}^p\). Apply this to the system \(\{x - 2 y + z = 0, \; 3 x + y - z = 0\}\) and exhibit a generating family of its solution space.
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