I have been given the following definition $V := \{ f : \mathbb{R} \to \mathbb{R} \mid \textrm{ there are } a_0,\ldots,a_4 \in \mathbb{R}$ and $f(x) = \sum_{i=0}^4 a_i x^i$ for all $x \in \mathbb{R}\}$. Furthermore the linear map $\varphi: V \to V$ is defined as $$\varphi(f)(x) = f''(x) - x \cdot f'(x) + f(x-1).$$
I want to prove that $\mathcal{B}:=\{1, x, x^2, x^3, x^4\}$ is a basis from $V$. I know it is very obvious that $\mathcal{B}$ is a basis of $V$ but I still need to prove it formally. I started, by trying to prove the linearity, so I came up with the following equation: $$a\cdot1 + b \cdot x + c\cdot x^2 + d\cdot x^3 + e \cdot x^4 = 0$$ for all $a,b,c,d,e \in \mathbb{R}$. But from this point on I am stuck. I don't know how I should continue, especially showing that $\mathcal{B}$ is a generating system of $V$.
The simplest thing to do is use Emory Sun's suggestion: "recall that a polynomial is 0 if and only if all its coefficients are zero". Of course proving that is essentially the same as proving the given statement. Recall that "$f(x)= a+ bx+ cx^2+ dx^3+ ex^4= 0$ means that it is 0 for all x, i.e. it is a constant. That is the basis for Monadologie's suggestion. Since it is a constant, $f'(x)= b+ 2cx+ 3dx^2+ 4e^3= 0$ for all x. And then $f''(x)= 2c+ 6dx+ 12ex^2= 0$, $f''(x)= 6d+ 24ex= 0$, and $f'''(x)= 242e= 0$. Setting x= 0 in each of those gives a= 0, b= 0, 2c= 0, 6d= 0, and 24e= 0.
A more "primitive" method (and more tedious method) is just to set x equal to 5 different values to get 5 equations to solve for a, b, c, d, and e: Taking x to be -2, -1, 0, 1, and 2 gives us a- 2b+ 4c- 8d+ 16e= 0, a- b+ c- d+ e= 0, a= 0, a+ b+ c+ d+ e= 0, and a+ 2b+ 4c+ 8d+ 16e= 0. Adding the first and fifth of those equation cancels b and d and leaves 2a+ 8c+ 32e= 0. Of course we know from the third equation that a= 0 so 8c+ 32e= 0 which reduces to c= -4e. Adding the second and fourth equations gives 2a+ 2c+ 2e= 0 which reduces to c= -e. c= -4e= -e so 5e= 0, e= 0 and then c= 0. If we were to subtract instead of adding equations, we would eliminate a, c, and e, leaving 4b+ 16d= 0 and 2b+ 2d= 0. Those reduce to b= 0 and d= 0.