Is the Set of Polynomials of Degree ≤ 6 with a3 = 3 a Vector Space?

[kq6up]

Homework Statement

For each of the following sets, either verify (as in Example 1) that it is a vector space, or show which requirements are not satisfied. If it is a vector space, find a basis and the dimension of the space.

6. Polynomials of degree ≤ 6 with a3 = 3.

Homework Equations

N/A

The Attempt at a Solution

I put that it was a vector space with the basis of {1,x,x^2,3*x^3,x^4,x^5,x^6} and dimension of 7. I am not sure why it fails to be a vector space.

Chris

 

[tms]

What happens if you add two members of the given set, or if you multiply a member by a scalar?

 

[kq6up]

I think there is a closure problem because one would not be able to get rid of the 3*x^3 by playing with coefficients.

I will have to think about it a little more.

Chris

 

[tms]

I think there is a closure problem $\ldots$

Definitely.