Why is any general cutset equation a linear combination of fundamental cutsets?

I have currently started studying circuit theory and my professor introduced graphs to form kvl and kcl equations. When we form the cutset matrix using fundamental cutsets it is easy to understand that equations corresponding to fundamental cutsets are linearly independent. But as soon as we add any cutset equation to the matrix which is not a fundamental cutset, the sum of entries of a column become zero signifying that the equation set is no longer linearly independent. My professor told us that each cutset can be written as a linear combination of fundamental cutsets and he showed this using an example. It wasn't quite satisfactory as no mathematical insight was given. Since I have just been introduced to graph theory, I couldn't come up with a formal proof of this. I looked up on the net and could not find any good material which adresses this. It would be very helpful if someone could explain why this is the case.

The same goes for fundamental loops that is every loop can be written as the linear combination of fundamental loops. It looks obvious but I still am not able to prove it.

• This might not be a complete answer, but consider whether the fundamental cutsets form a complete basis for some vector space. Jul 25 '19 at 14:27
• Elements of your sets form a vector field over the field $\{-1, 0, 1\}$. All vector fields have a basis when the Axiom of Choice is true. You can prove this by using Zorn’s Lemma. Jul 25 '19 at 15:04