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Write the polynomial as linear combinations of monic Legendre polynomials by using orthogonality to compute the coefficients.


My attempt:

Since I know that $q_4(t) = t^4-\frac{6t^2}{7}+\frac{3}{35}$ and $q_2(t)= t^2 -\frac{1}{3}$ then I must have a situation where the linear combinations looks like $$\alpha q_4(t) +\beta q_2(t) +\gamma q_0(t)$$where $\alpha , \beta, \gamma$ are constant coefficients. My question is how do I find those coefficients so that $$\alpha q_4(t) +\beta q_2(t) +\gamma q_0(t)= t^4 +t^2$$

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"My question is how do I find those coefficients": The question tells you how to find the coefficients. – wj32 Nov 2 '12 at 4:42
up vote 1 down vote accepted

You should actually write it out and gather like terms. $$\alpha q_4(t)+\beta q_2(t)+\gamma q_0(t)=\alpha t^4+\left(\beta-\frac67\alpha\right)t^2+\left(\frac3{35}\alpha-\frac13\beta+\gamma\right),$$ so we need $$\alpha=1,\quad\beta-\frac67\alpha=1,\quad\frac3{35}\alpha-\frac13\beta+\gamma=0.$$ Solve the system for $\alpha,\beta,\gamma.$

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ah of course, thank you very much Cameron !!! – diimension Nov 2 '12 at 4:50

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