Question

Using FTLM.

a) Let $q\in P_{2}(F)$. Use linear algebra to prove that there is a polynomial $p\in P_{2}(F)$ such that p + p' - 3p'' = q. Hint: consider the map defined by Tp: p + p' - 3p'', and use FTLM.

b) Let $x_{1}, x_{2}, x_{3}$ be distinct elements of $\mathbb{R}$. Let $y_{1}, y_{2}, y_{3}$ be any elements of $\mathbb{R}$. Use linear algebra to prove that there is a $p\in P_{2}(\mathbb{R})$ such that $p(x_{1}) = y_{1}, p(x_{2}) = y_{2}, p(x_{3}) = y_{3}.$ Hint: consider the map $S : P_{2}(\mathbb{R})\rightarrow \mathbb{R}^{3}$ defined by $Sp = (p(x_{1}), p(x_{2}), p(x_{3}))$ . You can use any facts from algebra about the solution set of a quadratic equation.

c) Prove that the p found in each part above is unique.

**FTLM is Fundamental Theorem of Linear Maps

Answer 1

"a) Let q E P_2 (F) Claim: - P elementof P_2 (F) s.t q = P + P' = 3P'' Let T: P_2 (F) Rightarrow P_2 (F), T(P) = P + P' = 3P'' T(1) = 1, T(x) = x + 1 T(x^2) = x^2 + 2x - 6 Therefore T has matrix form T(P) = A[P] A = [1 1 -6 0 1 2 0 0 1] of A is invertible Rightarrow dim Range A = 3 = dim P_2 (F) Therefore T is Onto for 2 E P_2 (F) T(P) = 9 i.e. P + P' = 3P' = 9 Hence Proved \r\n b) Let x_1 x_2 x_3 be distinct elements of P and y_1 y_2 y_3 elementof R Claim: P elementof P_2 (/R) S.t P(x_1) = y_1 P(x_2) = y_2, P(x_3) = y_3 define S: P_2 (R) Rightarrow R^3 S(P) =