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(Solution Library) Consider P_1 with the inner product < p, q>=∫_0^1 p(x) q(x) d x and basis \mathcalB=1+x, 1-x. Verify that the polynomials in \mathcalB are

Question: Consider \(P_{1}\) with the inner product

\[\langle p, q\rangle=\int_{0}^{1} p(x) q(x) \mathrm{d} x\]

and basis \(\mathcal{B}=\{1+x, 1-x\}\).

  1. Verify that the polynomials in \(\mathcal{B}\) are linearly independent.
  2. Use the Gram-Schmidt process to find polynomials with the same span as the vectors in \(\mathcal{B}\) that are othonormal with respect to the inner product (1).
  3. Now consider the basis \(\mathcal{C}=\{1, x\} .\) Show that
\[\langle p, q\rangle=[p]_{\mathcal{C}}^{t}\left(\begin{array}{cc} 1 & 1 / 2 \\ 1 / 2 & 1 / 3 \end{array}\right)[q]_{\mathcal{C}} \quad \text { for all } p, q \in P_{1}\]

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