The Puzzling Case of Positive Polynomials: A Vector Space?

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Homework Help Overview

The discussion revolves around whether the set of all polynomials with positive coefficients constitutes a vector space, focusing on the properties and conditions that define vector spaces.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the conditions for a vector space, questioning the closure under scalar multiplication and the existence of inverse elements. Some express confusion about the implications of specific examples.

Discussion Status

The conversation is ongoing, with participants examining the properties of vector spaces and how they apply to the set in question. There is a suggestion to specify the field for clarity, and some guidance is offered regarding the implications of scalar multiplication.

Contextual Notes

Participants note the importance of specifying the field, likely the real numbers, and discuss the implications of using negative scalars on polynomials with positive coefficients.

Workout
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Homework Statement


Is the set of all polynomials with positive coefficients a vector space?

It's not.

But after going through the vector space conditions I don't see how it can't be.
 
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what are the inverse elements of the space?
 
But after going through the vector space conditions I don't see how it can't be.
If you show your work, it is easier to find your error.
Oh, and you have to specify the field for the vector space (probably R).

Edit: Or use the direct hint given above ;).
 
One of the properties that a vector space must have is that it is "closed under scalar multiplication". That is, the product of any real number a, and vector v, av is also a number. v= x+ 1 is a "polynomial with positive coefficients" and a= -1 is a number. What is av?
 

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