Q4. Let R2 [x] be the vector space of polynomials of degree m is small or equal to 2. Consider the two following bases: the canonical E = {f1; x; x2}and a nest base B as follows B = {1; 1 + x; (1 + x) ^2}. Answer the following questions reasonably: (a) What are the coordinates in the base B of the vector p(x) = -x2 + 4? (b) And the coordinates in the canonical base E of the vector q (x) than in the base Bte coordinates (1; 1; 1) B? (c) Let U1 = {p(x) 2 R2[x] jp(0) = 0}į U2 = {p(x) 2 R2[x] j p(0) = 1} Are U1 and U2 vector spaces of R2(x]? Reason your answer, proving it in affirmative case.
Q4. Let R2 [x] be the vector space of polynomials of degree m is small or equal to 2. Consider the two following bases: the canonical E = {f1; x; x2}and a nest base B as follows B = {1; 1 + x; (1 + x) ^2}. Answer the following questions reasonably: (a) What are the coordinates in the base B of the vector p(x) = -x2 + 4? (b) And the coordinates in the canonical base E of the vector q (x) than in the base Bte coordinates (1; 1; 1) B? (c) Let U1 = {p(x) 2 R2[x] jp(0) = 0}į U2 = {p(x) 2 R2[x] j p(0) = 1} Are U1 and U2 vector spaces of R2(x]? Reason your answer, proving it in affirmative case.
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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![Q4.
Let R2 [x] be the vector space of polynomials of degree m is small or equal to 2. Consider the
two following bases: the canonical E = {f1; x; x2}and a nest base B as follows B =
{1; 1 + x; (1 + x) ^2}. Answer the following questions reasonably:
(a) What are the coordinates in the base B of the vector p(x) = -x2+ 4?
(b) And the coordinates in the canonical base E of the vector q (x) than in the base B te
coordinates (1; 1; 1) B?
(c) Let U1= {p(x) 2 R2[x] j p(0) = 0} į U2 = {p(x) 2 R2[x] j p(0) = 1} Are U1 and U2
vector spaces of R2(x]? Reason your answer, proving it in affirmative case.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e5b1a56-3032-449a-acc6-d2196fb84e38%2Fc95543a2-3ee5-4c98-8fd6-2cafce094ba5%2F3cis3c_processed.png&w=3840&q=75)
Transcribed Image Text:Q4.
Let R2 [x] be the vector space of polynomials of degree m is small or equal to 2. Consider the
two following bases: the canonical E = {f1; x; x2}and a nest base B as follows B =
{1; 1 + x; (1 + x) ^2}. Answer the following questions reasonably:
(a) What are the coordinates in the base B of the vector p(x) = -x2+ 4?
(b) And the coordinates in the canonical base E of the vector q (x) than in the base B te
coordinates (1; 1; 1) B?
(c) Let U1= {p(x) 2 R2[x] j p(0) = 0} į U2 = {p(x) 2 R2[x] j p(0) = 1} Are U1 and U2
vector spaces of R2(x]? Reason your answer, proving it in affirmative case.
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