Question: Suppose that \(W\) is a four dimensional vector space over a field \(\mathbb{F}\) with basis \(\mathcal{S}=\left(\mathbf{v}_{1}, \mathrm{v}_{2}, \mathbf{v}_{3}, \mathrm{v}_{4}\right)\)

1. Show that \(\mathcal{B}=\left(\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}, \mathbf{b}_{4}\right)\) is also a basis for \(W\), where
   \[\mathbf{b}_{1}=\mathbf{v}_{1}+2 \mathbf{v}_{3}+3 \mathbf{v}_{4}, \quad \mathbf{b}_{2}=\mathrm{v}_{2}+\mathbf{v}_{4}, \quad \mathbf{b}_{3}=\mathrm{v}_{1}+\mathrm{v}_{3}+2 \mathbf{v}_{4}, \quad \text { and } \quad \mathbf{b}_{4}=-\mathrm{v}_{2}+3 \mathbf{v}_{3}+\mathrm{v}_{4}\]
2. Find the change of basis matrix \(P\) from \(\mathcal{S}\) to \(\mathcal{B}\), so that
   \[[\mathrm{x}]_{\mathcal{B}}=P[\mathrm{x}]_{\mathcal{S}}, \quad \text { for all } \mathrm{x} \in W\]
3. Find the change of basis matrix \(Q\) from \(\mathcal{B}\) to \(\mathcal{S}\), so that
   \[[\mathrm{x}]_{\mathcal{S}}=Q[\mathrm{x}]_{\mathcal{B}}, \quad \text { for all } \mathrm{x} \in W\]
4. Calculate the product PQ.

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