Proof Technique: Linear Independence - What makes the technique work in general? Reading the book on Graph theory written by Bondy and Murty (Springer), they present the following proof technique (Linear Indepence) to use when the combinatorial approach fail.
My questions are:


*

*What makes the proof technique work on the specific case in the proof below ? They build a system of linear equations based on the decomposition of $K_n$ into complete bipartite graphs (CBG). Then they assume we have less than $n-1$ of these CBG's in the decomposition, which imply that there exist a non-trivial solution. And from this non-trivial solution we deduce a contradiction. However, I don't see where we use our decomposition in the proof other than constructing the system of equations in $n$ variables. Why couldn't I create this system in real-world and produce the contradiction ?

*In general what makes the proof technique tick ?

 A: 
However, I don't see where we use our decomposition in the proof other than constructing the system of equations in n variables. Why couldn't I create this system in real-world and produce the contradiction ?

It's a counting argument, so that's what makes it tick. So since $\mathcal{F}$ is a decomposition of $K_{n}$, we have the following:
$$
\bigcup_{i=1}^{k} E(F_{i}) = E(K_{n})
$$
Subject to $E(F_{i}) \cap E(F_{j}) =\emptyset$, $\forall{i, j}$ with $i \neq j$.
The idea is that if we have $n-2$ or fewer linear equations associated with $\mathcal{F}$, then we have extra edges and another partition we haven't considered. Look at $K_{4}$. Let's consider the decomposition into $K_{2, 2}$, $K_{2}$, and $K_{2}$. 
So since we are dealing with $K_{4}$, we are assuming at most $2$ subgraphs in the decomposition. I will chose the extremal cases: $K_{2,2}$ and $K_{2}$, which uses $5$ edges. However, we have a sixth edge (the second $K_{2}$), which we have not considered. Since $\mathcal{F}$ is a decomposition, that second $K_{2}$ is in $\mathcal{F}$. That's the contradiction underneath the linear equations.
