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Question Number 140176 by EDWIN88 last updated on 05/May/21

Let V be a vector space of polynomials p(x)= a+bx+cx^2 with real coefficients a,b and c. Define an inner product on V by (p,q)=(1/2)∫_(−1) ^1 p(x)q(x) dx . (a) Find a orthonormal basis for V consisting of polynomials φ_o (x) , φ_1 (x) and φ_2 (x) having degree 0,1 and 2 respectively.

$$\mathrm{Let}\:\mathrm{V}\:\mathrm{be}\:\mathrm{a}\:\mathrm{vector}\:\mathrm{space}\:\mathrm{of}\:\mathrm{polynomials} \\ $$$$\mathrm{p}\left(\mathrm{x}\right)=\:\mathrm{a}+\mathrm{bx}+\mathrm{cx}^{\mathrm{2}} \:\mathrm{with}\:\mathrm{real}\:\mathrm{coefficients} \\ $$$$\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}.\:\mathrm{Define}\:\mathrm{an}\:\mathrm{inner}\:\mathrm{product}\:\mathrm{on}\:\mathrm{V} \\ $$$$\mathrm{by}\:\left(\mathrm{p},\mathrm{q}\right)=\frac{\mathrm{1}}{\mathrm{2}}\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\mathrm{p}\left(\mathrm{x}\right)\mathrm{q}\left(\mathrm{x}\right)\:\mathrm{dx}\:. \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Find}\:\mathrm{a}\:\mathrm{orthonormal}\:\mathrm{basis}\:\mathrm{for}\:\mathrm{V}\:\mathrm{consisting} \\ $$$$\mathrm{of}\:\mathrm{polynomials}\:\phi_{\mathrm{o}} \left(\mathrm{x}\right)\:,\:\phi_{\mathrm{1}} \left(\mathrm{x}\right)\:\mathrm{and}\:\phi_{\mathrm{2}} \left(\mathrm{x}\right) \\ $$$$\mathrm{having}\:\mathrm{degree}\:\mathrm{0},\mathrm{1}\:\mathrm{and}\:\mathrm{2}\:\mathrm{respectively}. \\ $$$$ \\ $$

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