Let V = P (R) be equipped with the inner product (p, q) = Sp(x)q(x)dx and...

 

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Let V = P₂ (R) be equipped with the inner product (p, q) = Sp(x)q(x)dx and V→ R, be given by (p) = (-1)p+ (1)p, vp € V (a) Show that is a linear functional on V. (b) Given that if L. (x) = 12, 1, (x)=√x, L: (x) = 3√3 (x² - 4) then B = (Lo, L1, L2) is an orthonormal basis for Lo 1 12 V, compute the unique polynomial P 212 EV for which (p) = (q, P), Vq € V Let V = P₂ (R) be equipped with the inner product (p, q) = Sp(x)q(x)dx and V→ R, be given by (p) = (-1)p+ (1)p, vp € V (a) Show that is a linear functional on V. (b) Given that if L. (x) = 12, 1, (x)=√x, L: (x) = 3√3 (x² - 4) then B = (Lo, L1, L2) is an orthonormal basis for Lo 1 12 V, compute the unique polynomial P 212 EV for which (p) = (q, P), Vq € V

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a To show that is a linear functional on V we need to prove two properties additivity and homogeneit... View the full answer