# Vector Space Spanned by Legendre Polynomials

Problem
Let $V$ be the vector space over $\mathbb{R}$ spanned by Legendre Polynomials:

$P_0(x)=1$,$\space\space$$P_1(x)=x,\space\space$$P_2(x)=\frac{1}{2}(3x^2-1)$,\space\space$$P_3(x)=\frac{1}{2}(5x^3-3x) Consider the map H:V\rightarrow V, f\mapsto f^{\prime\prime}, where f^{\prime\prime} denotes the second derivative of f. You may assume H is a linear transformation. (a) Show that \beta:=\{P_0,P_1,P_2,P_3\} is a basis for V. (b) Determine the matrix representation of H with respect to basis \beta. Attempt (a) So getting the linear combination, p(x)=a_0+a_1x+a_2x^2+a_3x^3\in P_3 and finding co-efficients c_0, c_1, c_2, c_3, I ended up with:$$a_0=2c_0 - c_2a_1=2c_1-3c_3a_2=3c_2a_3=5c_3$$So when a_0=a_1=a_2=a_3=0, c_0=c_1=c_2=c_3=0. This shows the set spans V as it is linearly independent and therefore it is a basis. (b) I'm not sure how to do this part and I can't find any clear instructions for a question like this. Any help would be greatly appreciated. • For (b), you should consider writing the matrix representation of the second derivative operator with respect to the monomial basis first. – Chester Aug 13 '15 at 0:44 • P_{3}''=15x=15P_{1}, P_{2}''=3=3P_{0}, P_{1}''=0, P_{0}''=0. – DisintegratingByParts Aug 13 '15 at 3:58 ## 1 Answer Starting with$$ \begin{align} P_{0}'' & = 0P_{0}+0P_{1}+0P_{2}+0P_{3} \\ P_{1}'' & = 0P_{0}+0P_{1}+0P_{2}+0P_{3} \\ P_{2}'' = 3 & = 3P_{0}+0P_{1}+0P_{2}+0P_{3} \\ P_{3}'' = 15x & = 0P_{0}+15P_{1}+0P_{2}+0P_{3}, \end{align} $$which gives the following matrix representation of \frac{d^{2}}{dx^{2}}:$$\left[\begin{array}{cccc}0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 15 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right]$$This assumes that \alpha_0 P_0 + \alpha_1 P_1 + \alpha_2 P_2 + \alpha_3 P_3 is written as$$ \left[\begin{array}{c}\alpha_0 \\ \alpha_1 \\ \alpha_2 \\ \alpha_3\end{array}\right]\$

• I don't think I could have gotten a clearer explanation, thanks a million! – teme92 Aug 13 '15 at 8:26
• @teme92 : You're welcome. – DisintegratingByParts Aug 13 '15 at 10:28