finding the kernel of a polynomial transformation $P \to P$ • Let $P$ denote the vector space of polynomials and $l : P → P$ be defined via
$p(x) \mapsto x^2p(x) + xp'(x)$
How do I find the kernel of this transformation? I am having a hard time setting up a system that I can set equal to 0. Conceptually, I understand that if I can set up a matrix I will be able to show linear independence or dependence and from that determine the kernel.
 A: Setting $p(x)=0$ gets us $$x^2p(x)+xp'(x)=0$$
Factor out the x to get
$$x(xp(x)+p'(x))=0$$
since we want this to be the 0 polynomial, it has to be 0 for all $x$, not just $x=0$,  so we need
$$xp(x)+p'(x)=0$$
Solving this outside the context of polynomials would require differential equations.  However, since we know $p(x)$ is a polynomial,  we can avoid that difficulty.   Subtracting one term to the other side gets us
$$xp(x)=-p'(x)$$
This says the polynomial on the left is the same as the polynomial on the right, so they have the same degree.  If $p(x)=0$ (undefined degree),  the equation is true.
So assume $p(x)\neq 0$,  so it has a well defined degree.   What's the degree of the LHS and the RHS here? Can you finish?
A: If $p(x)=a_nx^n+a_{n-1}x^{n-1}+\dots +a_1x+a_0$, then $p'(x)=na_nx^{n-1}+\dots +a_1$.
Hence the condition is $(a_nx^{n+2}+a_{n-1}x^{n+1}+\dots+a_0x^2)+(na_nx^n+(n-1)a_{n-1}x^{n-1}+\dots+a_1x)=0\implies a_nx^{n+2}+a_{n-1}x^{n+1}+(a_{n-2}+na_n)x^n+\dots +(a_0+2a_2)x^2+a_1x=0$.
But when a polynomial equals zero, all of the coefficients are zero (reason? ).
So, $\forall i,2\le i\le n,\, a_{i-2}+ia_i=0$.  And $a_n=a_{n-1}=a_1=0$.
Thus the coefficients are all zero.
So the kernel is trivial.
