Dimension of certain linear space

Question

I have the linear space $V = \{ P \in \mathbb{R}_4[x] : xP'''(x) + P''(x) = P'(-1) + P(0) = 0 \}$ and have to find its dimension. In terms of tools I'm only allowed to use there's the theorem that for any linear $F : A \to B$ where $A$ and $B$ are linear spaces, it's true that $\dim (\ker (F)) + \dim (\mathrm{Im}(F)) = \dim (A)$, so my idea is to define some linear $F$ such that $F: \mathbb{R}_4[x] \to \mathbb{R}_4[x]$, but I have no idea for $F(P)$. $V$ looks like the kernel of some $F$, so it should work, but I don't know how to continue. Could you give me any hints?

Answer 1

You can consider two linear maps: \begin{align} F\colon \mathbb{R}_4[x] & \to \mathbb{R}_4[x], & F(P)&=xP'''(x)+P''(x) \\ G\colon \mathbb{R}_4[x] & \to \mathbb{R}, & G(P)&=P'(-1)+P(0) \end{align} Thus $V=\ker F\cap \ker G$.

The matrix of $F$ with respect to the basis $\{1,x,x^2,x^3,x^4\}$ is \begin{bmatrix} 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 12 & 0 \\ 0 & 0 & 0 & 0 & 36 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} and the matrix of $G$ is \begin{bmatrix} 1 & 1 & -2 & 3 & -4 \end{bmatrix} Can you finish?