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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Let R[X]n be the
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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