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

$$\mathrm{Let}\mathrm{V}\mathrm{be}\mathrm{a}\mathrm{vector}\mathrm{space}\mathrm{of}\mathrm{polynomials}$$$$\mathrm{p}\left(\mathrm{x}\right)=\mathrm{a}+\mathrm{bx}+{\mathrm{cx}}^2\mathrm{with}\mathrm{real}\mathrm{coefficients}$$$$\mathrm{a},\mathrm{b}\mathrm{and}\mathrm{c}.\mathrm{Define}\mathrm{an}\mathrm{inner}\mathrm{product}\mathrm{on}\mathrm{V}$$$$\mathrm{by}(\mathrm{p},\mathrm{q})=\frac{1}{2}\underset{-1}{\stackrel{1}{\int }}\mathrm{p}\left(\mathrm{x}\right)\mathrm{q}\left(\mathrm{x}\right)\mathrm{dx}.$$$$\left(\mathrm{a}\right)\mathrm{Find}\mathrm{a}\mathrm{orthonormal}\mathrm{basis}\mathrm{for}\mathrm{V}\mathrm{consisting}$$$$\mathrm{of}\mathrm{polynomials}{\varphi }_{\mathrm{o}}\left(\mathrm{x}\right),{\varphi }_1\left(\mathrm{x}\right)\mathrm{and}{\varphi }_2\left(\mathrm{x}\right)$$$$\mathrm{having}\mathrm{degree}0,1\mathrm{and}2\mathrm{respectively}.$$$$$$

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