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(a) Let Pn(R) denote the vector space of polynomials of degree at most n, where addition is the usual addition of functions, and scalar multiplication is the usual way we multiply a function by a real number. Consider the map T: P3(R)→ P₂ (R), given by T(p(x)) = 3p" (r) - 2p'(x). Find a basis B for P3(R) and a basis B' for P2 (R) (and explain why these are bases). Compute the (B, B')-matrix for T. (b) Prove that if p is a non-constant polynomial with degree at most 3, then p can not be a solution to the differential equation 3p" (x) - 2p'(x) = 0. (The reason for "non-constant" above is that of course any constant function will satisfy this because both its derivative and its second derivative will be 0).

(a) Let Pn(R) denote the vector space of polynomials of degree at most n, where addition is the usual addition of functions, and scalar multiplication is the usual way we multiply a function by a real number. Consider the map T: P3(R)→ P₂ (R), given by T(p(x)) = 3p" (r) - 2p'(x). Find a basis B for P3(R) and a basis B' for P2 (R) (and explain why these are bases). Compute the (B, B')-matrix for T. (b) Prove that if p is a non-constant polynomial with degree at most 3, then p can not be a solution to the differential equation 3p" (x) - 2p'(x) = 0. (The reason for "non-constant" above is that of course any constant function will satisfy this because both its derivative and its second derivative will be 0).

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.3: Orthonormal Bases:gram-schmidt Process
Problem 17E: Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner...
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5. Let Pn (R) denote the vector space of polynomials of degree at most n, where addition is the usual addition of functions, and scalar multiplication is the usual way we multiply a function by a real number. Consider the map T: P3(R) → P₂ (R), given by T(p(x)) = 3p" (r) - 2p'(x). Find a basis B for P3(R) and a basis B' for P2 (R) (and explain why these are bases). Compute the (B, B')-matrix for T. (b) Prove that if p is a non-constant polynomial with degree at most 3, then p can not be a solution to the differential equation 3p" (x) - 2p'(x) = 0. (The reason for "non-constant" above is that of course any constant function will satisfy this because both its derivative and its second derivative will be 0).
Transcribed Image Text:5. Let Pn (R) denote the vector space of polynomials of degree at most n, where addition is the usual addition of functions, and scalar multiplication is the usual way we multiply a function by a real number. Consider the map T: P3(R) → P₂ (R), given by T(p(x)) = 3p" (r) - 2p'(x). Find a basis B for P3(R) and a basis B' for P2 (R) (and explain why these are bases). Compute the (B, B')-matrix for T. (b) Prove that if p is a non-constant polynomial with degree at most 3, then p can not be a solution to the differential equation 3p" (x) - 2p'(x) = 0. (The reason for "non-constant" above is that of course any constant function will satisfy this because both its derivative and its second derivative will be 0).
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