Let R[X]n be the vector space of polynomials with degree ≤ n. Define T : R[X]n → R[X]n by T : p(X) → −p''(X) + Xp'(X). 1. Prove that T indeed maps R[X]n to itself and is linear. 2. Show that 1, X, X2, . . . , Xn is a basis of R[X]n. 3. Find all eigenvalues of T in the case where n = 2. 4. Finds all eigenvalues in the general case. Please do it step by step in detail, show why and how you take certain steps. Somehow questions with matrix go smoothly but when we deviate from that IM clueless

Let R[X]n be the vector space of polynomials with degree ≤ n. Define T : R[X]n → R[X]n by T : p(X) → −p''(X) + Xp'(X). 1. Prove that T indeed maps R[X]n to itself and is linear. 2. Show that 1, X, X2, . . . , Xn is a basis of R[X]n. 3. Find all eigenvalues of T in the case where n = 2. 4. Finds all eigenvalues in the general case. Please do it step by step in detail, show why and how you take certain steps. Somehow questions with matrix go smoothly but when we deviate from that IM clueless

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let R[X]_n be the vector space of polynomials with degree ≤ n.
Define T : R[X]_n → R[X]_n by T : p(X) → −p''(X) + Xp'(X).
1. Prove that T indeed maps R[X]_n to itself and is linear.
2. Show that 1, X, X², . . . , Xⁿ is a basis of R[X]_n.
3. Find all eigenvalues of T in the case where n = 2.
4. Finds all eigenvalues in the general case.

Please do it step by step in detail, show why and how you take certain steps. Somehow questions with matrix go smoothly but when we deviate from that IM clueless

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