Dimension of the kernel of a differentiation linear transformation 
The set of all polynomials in a single variable $x$ forms a vector space $P$ of infinite dimension. Differentiation is a
  linear transformation on this vector space: 
$\frac{d}{dx}: P → P, p(x) → p'(x)$.
(a) What is the dimension of the kernel of $\frac{d}{dx}$
  as a linear transformation on $P$ ? 
(b) The linear transformation $\frac{d}{dx}
+ 2x$ acts on $P$ as $p(x) → p'(x) + 2xp(x)$. What is the dimension of its kernel?

I do know what dimensions and kernels of matrices are but this question is confusing me and I don't really understand it. Would really appreciate some help.
 A: (a) means "Which polynomials $p$ have the property that $p'=0$?" and, considering the fact that the collection of all such $p$ will form a vector space, "What is the dimension of the collection of such polynomials?"
(Just remember back to basic calculus - what kind of function has derivative $0$?)
(b) means "Which polynomials $p$ have the property that $p'+2xp=0$?" and, considering the fact that the collection of all such $p$ will form a vector space, "What is the dimension of the collection of such polynomials?"
In each case, you have a differential equation to solve, and then you must determine the dimension of the collection of solutions.
A: The null vector for $F[P(x)]$ is simply $0$, and so both (a) and (b) are simply asking how many independant vectors map to zero using the transformations $\frac{d}{dx}$ and $\frac{d}{dx}+2x$ respectively. You can solve this by setting up the simple differential equations
(a) $\frac{d}{dx}p(x)=0$ 
and
(b) $\frac{d}{dx}p(x)+2xp(x)=0$
and then solving using your knowledge of basic calculus, note that the kernel can simply be the null vector itself, also note that the kernel always contains the null vector.
It may also help to remember that any for any constant c, $c=cx^0$
