Linear Algebra, linear maps, kernels, and images Consider the map $L\colon P_n(\mathbb{R}) \rightarrow P_n(\mathbb{R})$, $L(P(x)) = P(x) - xP'(x),$ where $P_n(\mathbb{R})$ is the vector space of all polynomials of degree at most n with coefficients in $\mathbb{R}$.
a) Determine the kernel of $L$. What is its dimension?
b) Determine a basis for the image of $L$.
I'm not confident in getting the kernel. I know I need $L(v) = 0$, but I always feel like I'm missing something. Any help is appreciated.

A visual way to do that is to compute the image of the canonical basis $\{1,x,\ldots,x^n\}$.
This is $\{1,0,-x^2,-2x^3,\ldots,-(n-1)x^n\}$.
If you want, now you can write down the matrix of $L$ with respect to this basis.
Either way, you should easily see that 
$$
\mbox{Ker }L=\mathbb{R}x \quad \mbox{and}\quad \mbox{Im }L=\mbox{span}\{1,x^2,\ldots,x^n\}.
$$
