1.9.5 Linear spaces

Problem: Let \( V \) - linear space of all symmetric polynomials of degree at most two over \( \mathbb{R} \) from two variables \( x \) and \( y \). Choose a basis in space \( V \) and find the operator matrix \( L \) in this basis, if \[ L(f)(x, y)=(2 x+3 y) \frac{\partial f}{\partial x}+(3 x+2 y) \frac{\partial f}{\partial y} . \]