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.

You do indeed need all polynomials $p(x)$ such that $L(p(x))=0$, that is, $p(x)-xp^\prime(x)=0$.
If we denote $p(x)=a_0+a_1x+\ldots+a_nx^n$, then $xp^\prime(x)=a_1x+2a_2x^2+\ldots+ka_kx^k+\ldots+na_nx^n$.
To get $p(x)-xp^\prime(x)=0$, you need that for all $k\in\{0,\ldots,n\}$, $a_k-ka_k=0$, or in other words, $(k-1)a_k=0$, so for all $k\ne1$, $a_k=0$, and $a_1$ can be whatever you want.
So $\ker L = \{ax | a\in \mathbb{R}\}$.
