1.W.4 We'll work inside the vector space of polynomials in degree < 2, which is denoted P<2. Let P1 = 1, p2 = x + 2, and p3 = (x + 2)². Leť's think about Span{p1, P2, P3}. a) What polynomial is 3p1 + 2p2 – P3? b) I claim that a = a¡P1 + a2P2 + azp3 for some coefficients a1, a2, az in R. Find a1, a2, az. Hint: az = 0. c) I claim that a? = bịPi + b2p2 + b3p3 for some coefficients b1, b2, bz in R. Find b1, b2, bz. Hint: This problem isn't quite so cut and dry. Try to find three equations, one for each coefficient in the polynomial, and solve them for b1, b2, b3.
1.W.4 We'll work inside the vector space of polynomials in degree < 2, which is denoted P<2. Let P1 = 1, p2 = x + 2, and p3 = (x + 2)². Leť's think about Span{p1, P2, P3}. a) What polynomial is 3p1 + 2p2 – P3? b) I claim that a = a¡P1 + a2P2 + azp3 for some coefficients a1, a2, az in R. Find a1, a2, az. Hint: az = 0. c) I claim that a? = bịPi + b2p2 + b3p3 for some coefficients b1, b2, bz in R. Find b1, b2, bz. Hint: This problem isn't quite so cut and dry. Try to find three equations, one for each coefficient in the polynomial, and solve them for b1, b2, b3.
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:1.W.4 We'll work inside the vector space of polynomials in degree < 2, which is denoted P<2. Let
P1 = 1, p2 = x + 2, and p3 = (x + 2)². Leť's think about Span{p1, P2, P3}.
a) What polynomial is 3p1 + 2p2 – P3?
b) I claim that a = a¡P1 + a2P2 + azp3 for some coefficients a1, a2, az in R. Find a1, a2, az. Hint:
az = 0.
c) I claim that a? = bịPi + b2p2 + b3p3 for some coefficients b1, b2, bz in R. Find b1, b2, bz. Hint:
This problem isn't quite so cut and dry. Try to find three equations, one for each coefficient
in the polynomial, and solve them for b1, b2, b3.
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