Let $V$ denote the vector space of polynomials $f(x, y)$ in two variables of total degree at most $n$. Find the dimension of $V$.

If $S: V \rightarrow V$ is defined by

$$(S f)(x, y)=x^{2} \frac{\partial^{2} f}{\partial x^{2}}+y^{2} \frac{\partial^{2} f}{\partial y^{2}}$$

find the kernel of $S$ and the image of $S$. Compute the trace of $S$ for each $n$ with $1 \leqslant n \leqslant 4$.