Finding matrices representing the linear transformations

Question

Problem:

Let C₀ ={SUM (i = 1 to p) εivi¦εi is an element of F₂, vi is an element of V(G)}
be the vector space of 0-chains
and
Let C₁ ={SUM (i = 1 to q) εiei¦εi is an element of F₂, ei is an element of E(G)}
be the vector space of 1-chains

Recall the linear transformations
Boundary ∂ : C₁ → C0 defined by ∂(uv) = u + v and
Coboundary δ: C0 → C1 defined by δ(u) = SUM ei, where ei is adjacent to v.

Let Z(G) = { x an element of C1¦∂(x) = 0} be the cycle space of G
and
Let B(G) ={ x an element of C1¦there exists y an element of C0, x = ∂(y)} be the coboundary space of G.

a. Find matrices representing the linear transformations ∂ and δ.

b. Define an inner product on C₁ by < x,y > = SUM εiηi, where x = SUM εiei and y = SUM ηiei.
Prove that x is an element of Z(G) iff < x,y > = 0 for all y element of B(G)

c. Show that the dimensions of B(G) is p – k(G).

d.Characterize the class of graphs for which B(G) = C₁(G)