I have the linear transformation $L_1: V \rightarrow V$ (where $V = \mathbb{C}[x,y]$ is the vector space of polynomials with complex coefficients) defined as:
$L_1(f(x,y)) = \frac{f(x,y)+f(y,x)}{2}$
I have found the eigenspaces for $L_1$ as $E_0$ = $ker(L_1) = \{f(x,y) \in V:f(x,y)=-f(y,x)\}$ and $E_1 = im(L_1) = \{f(x,y) \in V: f(x,y)=f(y,x)\}$. Now I want to find a basis for these vector spaces. I tried writing out all the elements in each space, for example for $E_0$: $\{x-y, x^2 - y^2, x^3-y^3, ...\}$. I think this is a basis for $E_0$ but I am not sure if this is the best approach.
Thanks!
You are thinking right, but you are missing the monomials of the type $x^ky^l$, with $k,l\geqslant0$. Actually, a basis of $E_0$ is the set of all polynomials of the type $x^ky^l-x^ly^k$, with $k>l\geqslant0$. And a basis of $E_1$ is the set of all polynomals $x^ky^l+x^ly^k$ with $k\geqslant l\geqslant0$.