Dimension of the vector space of odd polynomials of degree $n$

Question

I was doing some exercises of vector space. Don't know how to handle this question?

A polynomial $p(x)$ is odd if $p(-x)=-p(x)$. Let $O_n$ denote the sets of odd polynomials in $P_n$. Then what is $\dim\,O_n$?

Answer 1

We can split functions into odd and even parts via $$f_{\mathrm{odd}}(x)=\frac{f(x)-f(-x)}{2},\qquad f_{\mathrm{even}}(x)=\frac{f(x)+f(-x)}{2}.$$

1: Check that these are odd and even respectively, and that $f=f_{\mathrm{odd}}+f_{\mathrm{even}}$.

2: Check that $(f+g)_{\mathrm{odd}}=f_{\mathrm{odd}}+g_{\mathrm{odd}}$ and $(f+g)_{\mathrm{even}}=f_{\mathrm{even}}+g_{\mathrm{even}}$.

3: Show that $f$ is odd (resp. even) if and only if $f_{\mathrm{even}}$ (resp. $f_{\mathrm{odd}}$) is identically zero.

4: Observe that even powers $x^{2k}$ are even and odd powers $x^{2k-1}$ are odd.

5: Explain why $f\in P_n$ is odd (resp. even) if and only if it contains no even (resp. odd) powers.

6: Observe that $\{x,x^3,x^5,\dots,x^m\}$ is a basis for $O_n $, where $m$ is the largest odd $\le n$.

Now how many elements are in this set?

Answer 2

Remember, a polynomial P(x) is odd iff it only contains odd powers of x. So every odd polynomial is of the form:

$a_1x^1 + a_3x^3 + \cdots + a_kx^k$

Assuming $P_n$ means the vector space of polynomials of degree n, you can figure out the number of different coefficients $a_i$ that will show up for a given value of n without too much trouble. That number will be the number of "degrees of freedom" you have in choosing an odd polynomial (equivalently, the number of basis elements), so it will be the dimension of the vector space.