Question: Let \(P_{2}\) be the vector space of all polynomials with degree at most 2 , and \(\mathcal{B}\) be the basis \(\left\{1, x, x^{2}\right\}\). Consider the linear operator \(T: P_{2} \rightarrow P_{2}\) given by \(T(p(x))=p(2 x+1)\); thus

1. Find \(T(1), T(x)\), and \(T\left(x^{2}\right)\), and compute the matrix \([T]_{\mathcal{B}, \mathcal{B}}\) for \(T\) with respect to the basis \(\mathcal{B}\)
2. Use the matrix to compute \(T\left(3+x+2 x^{2}\right)\)
3. Compute \(T\left(3+x+2 x^{2}\right)\) directly and compare with your answer to the previous part
4. What is the matrix for \(T \circ T\) with respect to the basis \(\mathcal{B}\) ?
5. Now let \(\mathcal{C}=\left\{1+x, 1+x^{2}, x+x^{2}\right\}\), and find the transition matrix from \(\mathcal{B}\) to \(\mathcal{C}\). Using this matrix, compute \([T]_{\mathcal{C}, \mathcal{C}}\).
6. Check your answer to the previous part by computing \([T]_{c, c}\) directly.

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