I have trouble finding a basis for $Map(\mathbb{R},\mathbb{R})$. While I know what a/the (standard)basis is for $\mathbb{R}^{2\times1}$, that should be $\left(\begin{array}{c} 1 \\ 0 \end{array}\right),\left(\begin{array}{c} 0 \\ 1 \end{array}\right) $ I can't seem to grasp how exactly a basis for $Map(\mathbb{R},\mathbb{R})$ could look like.
I have the following Vectorspace $V := \{ f : \mathbb{R} \to \mathbb{R} \mid \textrm{ there exists } a_0,\ldots,a_4 \in \mathbb{R}$ and $f(x) = \sum_{i=0}^4 a_i x^i$ for all $ x \in \mathbb{R}\}$ and there is a linear Map $\varphi: V \to V$ with $\varphi(f)(x) = f''(x) - x \cdot f'(x) + f(x-1).$
I have problems with the following two questions:
1) Calculate the matrix $M_{\mathcal{B}}^{\mathcal{B}}(\varphi)$ with respect to a base $\mathcal{B}$ of $V$.
2) Find a basis for $ker(\varphi)$
Now how does a base B for $Map(\mathbb{R},\mathbb{R})$ look like? Is there something like a canonic basis for $Map(\mathbb{R},\mathbb{R})$? I'm stuck and I'd be grateful for any hint into the right direction!
Could I assume that there is some $e$ which is the canonic base for $Map(\mathbb{R},\mathbb{R})$ such that $M_{\mathcal{B}}^{\mathcal{B}}(\varphi) = e*(f''(x) - x \cdot f'(x) + f(x-1)) $?
For clarity: $M_{\mathcal{B}}^{\mathcal{B}}(\varphi)$ is the Transformation matrix.