1. Consider the set V of all polynomials of degree two or less, along with the following definitions of addition and scalar multiplication: (azx? + a,x + a,) + (bzx? + b,x + bo) = (a, + bo)x? + (a, + b,)x+ (ao + bo) k(azx? + a,x + ao) = ka,x? + ka,x + kaz a. Is Vector Space Axiom 7, k(ü + v) = kū + kö satisfied? Support your answer appropriately. b. Would 0 0x2 + 0x +0 be the zero vector used to verify Axiom 4, ü +0 =0+ ü ? Explain. c. Given your answer to part b, is it possible to verify Axiom 5, u+-ü = -u+u = 0 ? If yes, define-ū and verify the axiom. If no, explain why. %3D %3D
1. Consider the set V of all polynomials of degree two or less, along with the following definitions of addition and scalar multiplication: (azx? + a,x + a,) + (bzx? + b,x + bo) = (a, + bo)x? + (a, + b,)x+ (ao + bo) k(azx? + a,x + ao) = ka,x? + ka,x + kaz a. Is Vector Space Axiom 7, k(ü + v) = kū + kö satisfied? Support your answer appropriately. b. Would 0 0x2 + 0x +0 be the zero vector used to verify Axiom 4, ü +0 =0+ ü ? Explain. c. Given your answer to part b, is it possible to verify Axiom 5, u+-ü = -u+u = 0 ? If yes, define-ū and verify the axiom. If no, explain why. %3D %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
plz provide answer q1
![1.
two or less, along with the following definitions of addition
and scalar multiplication:
Consider the set V of all polynomials of degree
(azx? + a,x + a,) + (bzx2 + b,x + bo)
= (ao + bo)x2 + (az + b,)x + (ao + bo)
k(azx? + a,x + ao) = ka,x? + ka,x + kaz
%3D
a. Is Vector Space Axiom 7, k(ü + i) = kū + ki
satisfied? Support your answer appropriately.
b. Would 0 = 0x2 + 0x +0 be the zero vector used to
verify Axiom 4, ü +0 =0+i ü ? Explain.
c. Given your answer to part b, is it possible to verify
Axiom 5, u+-u = -u+u 0 ? If yes, define -ü
and verify the axiom. If no, explain why.
%3D
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40108b0a-265e-408e-965f-e28d3d5aaedb%2F0382158b-7f2e-40e2-84d0-4032664c048e%2Fr20ef7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.
two or less, along with the following definitions of addition
and scalar multiplication:
Consider the set V of all polynomials of degree
(azx? + a,x + a,) + (bzx2 + b,x + bo)
= (ao + bo)x2 + (az + b,)x + (ao + bo)
k(azx? + a,x + ao) = ka,x? + ka,x + kaz
%3D
a. Is Vector Space Axiom 7, k(ü + i) = kū + ki
satisfied? Support your answer appropriately.
b. Would 0 = 0x2 + 0x +0 be the zero vector used to
verify Axiom 4, ü +0 =0+i ü ? Explain.
c. Given your answer to part b, is it possible to verify
Axiom 5, u+-u = -u+u 0 ? If yes, define -ü
and verify the axiom. If no, explain why.
%3D
%3D
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