4.1.18. Let V = P^{(1)} be the vector space consisting of linear polynomials p(t) = at + b. (a) Carefully explain why ?p, q? = ?_0^1 t p(t) q(t) dt defines an inner product on V. (b) Find all polynomials p(t) = at + b ? V that are orthogonal to p_1(t) = 1 based on this inner product. (c) Use part (b) to construct an orthonormal basis of V for this inner product. (d) Find an orthonormal basis of the space P^{(2)} of quadratic polynomials for the same inner product. Hint: First find a quadratic polynomial that is orthogonal to the basis you constructed in part (c).
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(a) To show that $(p, q) = \int_{0}^{1} p(t)q(t) dt$ defines an inner product on $V$, we need to verify the following properties: Show more…
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