adjacency operator on real functions of graph vertex set

Question

I am reading an article, and it says:
Let

1. $G$: a graph with vertex set $V$
2. $i\sim j$: vertex $i$ adjacent to vertex $j$
3. $F(V)$: the space of real functions on $V$
4. $e_i$: the characteristic vector of the vertex $i$ of $G$
5. the vectors $e_i$ form the standard basis of $F(V)$

Now, define the adjacency operator $\mathcal{A}$ being the linear operator on $F(V)$: $$(\mathcal{A}f)(i) = \sum_{j\sim i} f(j)$$

The matrix that represents $\mathcal{A}$ relative to the standard basis is the adjacency matrix of $G$.

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I want to come up with an example to see this. Consider a path graph with three nodes. The adjacency matrix is: $$A = \begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix}$$ Suppose $i = 2$; hence, $j = 1,3$. But I have no idea how to come up with a function $f\in F(V)$. What does $f(i)$ look like?

Note: I think for example $e_1 = [1,0,0,\ldots,0]$

Thanks!

Answer 1

Think of $f$ as a vector in $\mathbb{R}^{|V|}$, where $|V|$ is the number of vertices in $G$. Then the article uses $f(i)$ to denote the $i$th entry of $f$.