I am currently taking an introduction to graph theory course. We are using the 5th edition book by Robin J Wilson. At the end of chapter 1 there are some challenging problems I'd like to think about. But the problem is I don't exactly know what it is asking me. Here is the problem:
If $G$ is a simple graph with the edge-set $E(G)$, the vector space of G is the vector space over the field $\mathbb{Z}_2 = \{0,1\}$ of integers mod 2, whose elements are subsets of $E(G)$. The sum $E+F$ of two such subset $E$ and $F$ is the set of edges in $E$ or $F$ but not both and scaler multiplication is defined by $1\cdot E = E$ and $0\cdot E = \text{empty set}$. Show that this defines a vector spacee over $\mathbb{Z}_2$ and find a basis for it
So I've taken linear algebra, abstract algebra so I am familiar with what a vector field is. Similarly I also know what a field is and I also know simple modular arithmetic. However, I dont know how to combine everything together. If M.SE could be kind enough to go through the problem and explain what each part is doing and the meaning and possibly a solution. For example what does it mean by "vector space of G over the field.."