Question: Let \(P_{2}\) denote the real vector space of all polynomials in \(x\) with real coefficients and degree at most 2 . Define two transformations \(S: P_{2} \rightarrow P_{2}\) and \(T: P_{2} \rightarrow P_{2}\) by

1. Find the compositions \(S \circ T(p(x))\) and \(T \circ S(p(x))\).
2. Let \(\mathscr{B}=\left(1, x, x^{2}\right)\) denote the standard basis of \(P_{2}\). For each \(0 \leq k \leq 2\) calculate the coordinates \(\left[S\left(x^{k}\right)\right]_{\mathscr{B}}\) and \(\left[T\left(x^{k}\right)\right]_{\mathscr{B}}\)
3. Find matrices \(M\) and \(N\) so that \([S(p(x))]_{\mathscr{B}}=M[p(x)]_{\mathscr{B}}\) and \([T(p(x))]_{\mathscr{B}}=N[p(x)]_{\mathscr{B}}\) for all \(p(x) \in P_{2}\).

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