Is the set of polynomials with odd degree a subspace of $V$, the space of all nth degree polynomials?

Question

Let $V$ be a vector space of all $n$th degree polynomials.

Then the question is: Is the set of polynomials with odd degree a subspace of $V$?

Answer 1

Although late, but like orole hinted at, since $0$ is considered a polynomial of even power, the subset is not a subspace. A second reason that it is not a subspace is the fact that it is not closed under addition. Take for example $p_1 = 4 + 3x +x^2 + 3x^3$ and $p_2 = 2 - 5x +3x^2 - 3x^3$. Now notice that $p_1 + p_2 = 6 -2x + 4x^2$, which is not of odd degree.

Also, I would recommend you change the wording of $V$ to be defined as the vector space of all polynomials up to degree $n$ so that it isn't mistaken to be the polynomials of exactly degree $n$, which is not a vector space for the same reasons mentioned above.

Answer 2

No, it can not become a subspace. As usual, you say the zero element of the vector space and its subspaces is same. But the zero polynomial has a degree zero (which is even).

Another approach you can see, $f(x)=x^3+x^2$ and $g(x)=-x^3$, both are odd degree polynomial but $f(x)+g(x)=x^2$, which is of even degree, so vector addition fails. Hence set of odd degree polynomials can not become a subspace.