Graph theory (incidence matrix and linear algebra)

[TheMathNoob]

Homework Statement

I can't understand this paper. I understand the whole incidence matrix stuff, but I don't quiet get how it relates to the linear algebra. I don't know if this is allowed to do, but I will ask you questions line by line, so basically you will read the paper with me explaining every single detail if it's possible.

The first things that I would like to understand are the following transformations.
C1(Γ) ≅ R m; coordinates φ ↦ φ(ea) and
C0(Γ) ≅ R n ; coordinates f ↦ f(vi).

what I understand about this is that a function takes on vertices and outputs something that relates to edges and viceversa. But this is too vague. I want to know more. If it's possible can you relate that to what they are trying to do in the paper?.

Homework Equations

The Attempt at a Solution

 

[haruspex]

The paper says "recall that", implying there was some earlier discussion of these C functions. I'm not able to guess what they are.
By the way, I think the title is supposed to say "kernel and co-kernel", not "kernel and cockerel"