Thread: Question about proof of subspace.

Ok I want to make sure I am understanding this question and the concepts behind it.

*I'm going to use P,4 to denote P subscript 4*

Q: Determine whether the following are subspaces of P,4 (be careful!):

a) the set of polynomials in P,4 of even degree

b) the set of all polynomials of degree 3

c) the set of all polynomials p(x) in P,4 such that p(0) = 0

d) the set of all polynomials in P,4 having at least one real root.


A: Ok so my thinking is to show an example of say 5x^4 + 2x^3 + x^2 + x + 15
and add it to -5x^4 + 3x^3 + 4x^2 + 3x + 14 which makes a polynomial of degree 3 which does not exist in this subset. Thus, it does not satisfy the closure property.

Is this answering what the question is asking? Am I understanding the concepts correctly? Any help would be greatly appreciated.

Originally Posted by billbarber

Ok I want to make sure I am understanding this question and the concepts behind it.

*I'm going to use P,4 to denote P subscript 4*

Q: Determine whether the following are subspaces of P,4 (be careful!):

a) the set of polynomials in P,4 of even degree

b) the set of all polynomials of degree 3

c) the set of all polynomials p(x) in P,4 such that p(0) = 0

d) the set of all polynomials in P,4 having at least one real root.


A: Ok so my thinking is to show an example of say 5x^4 + 2x^3 + x^2 + x + 15
and add it to -5x^4 + 3x^3 + 4x^2 + 3x + 14 which makes a polynomial of degree 3 which does not exist in this subset. Thus, it does not satisfy the closure property.

Is this answering what the question is asking? Am I understanding the concepts correctly? Any help would be greatly appreciated.

Yes, that is exactly right.