Consider the set W of all polynomials of the form p = a0 + ajx + a2x² where ao, a1, and az are integers. Demonstrate one of the axioms for the definition of a vector space that does not hold to confırm that this is not a vector space under the usual polynomial addition and scalar multiplication Be clear which axiom (give the axiom, not just the number from the list) and show clearly why that axiom fails. Note: One way to show that an axiom fails is to find a counter example. That is a specific vector p (and other polynomial vectors if more than one vector in the axiom) and specific scalar(s) to show that the left side of the axiom does not equal to the right side.
Consider the set W of all polynomials of the form p = a0 + ajx + a2x² where ao, a1, and az are integers. Demonstrate one of the axioms for the definition of a vector space that does not hold to confırm that this is not a vector space under the usual polynomial addition and scalar multiplication Be clear which axiom (give the axiom, not just the number from the list) and show clearly why that axiom fails. Note: One way to show that an axiom fails is to find a counter example. That is a specific vector p (and other polynomial vectors if more than one vector in the axiom) and specific scalar(s) to show that the left side of the axiom does not equal to the right side.
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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![Consider the set **W** of all polynomials of the form
\[ p = a_0 + a_1 x + a_2 x^2 \]
where \( a_0 \), \( a_1 \), and \( a_2 \) are integers.
Demonstrate one of the axioms for the definition of a vector space that does not hold to confirm that this is not a vector space under the usual polynomial addition and scalar multiplication.
**Be clear which axiom (give the axiom, not just the number from the list) and show clearly why that axiom fails.**
**Note:** One way to show that an axiom fails is to find a counterexample. That is, find a specific vector \( p \) (and other polynomial vectors if more than one vector is included in the axiom) and specific scalar(s) to show that the left side of the axiom does not equal the right side.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd2082cb3-95cf-4561-a9bf-97e35996c7c7%2F84071364-1c48-4837-b3bc-0bfcaaab0cc6%2Fc3zxlc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the set **W** of all polynomials of the form
\[ p = a_0 + a_1 x + a_2 x^2 \]
where \( a_0 \), \( a_1 \), and \( a_2 \) are integers.
Demonstrate one of the axioms for the definition of a vector space that does not hold to confirm that this is not a vector space under the usual polynomial addition and scalar multiplication.
**Be clear which axiom (give the axiom, not just the number from the list) and show clearly why that axiom fails.**
**Note:** One way to show that an axiom fails is to find a counterexample. That is, find a specific vector \( p \) (and other polynomial vectors if more than one vector is included in the axiom) and specific scalar(s) to show that the left side of the axiom does not equal the right side.
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