Advanced Engineering Mathematics
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ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.](https://content.bartleby.com/qna-images/question/d2082cb3-95cf-4561-a9bf-97e35996c7c7/84071364-1c48-4837-b3bc-0bfcaaab0cc6/c3zxlc_processed.jpeg)
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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