(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).
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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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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