A standard application of Linear Algebra is circuits and Kirchhoff's Laws. Does anyone know of a proof of uniqueness of a solution of a system given by these laws? There are many, many examples, but little theory regarding why there is always a unique solution.
For reference (Wikipedia)
At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node
The algebraic sum of the products of the resistances of the conductors and the currents in them in a closed loop is equal to the total emf available in that loop.
My thoughts are as follows: Initially these laws setup two systems $Ax = 0$ and $Bx= b$, respectively. If there are $n$ nodes and m currents, then $A$ is a $n \times m$ matrix. If there are l loops in the circuit, then $B$ is a $l \times m$ matrix. I tried to work with the augmented matrix $$ \left[\begin{matrix} A \\B\end{matrix} \right|\left.\begin{matrix}0\\b\end{matrix}\right]$$ But I see no reason why this always has a unique solution.
There is a statement that works for your purposes (proposition 9.4) presented in Markov Chains and Mixing Times (Levin, Peres, Wilmer) that you can follow by reading pages 115-118.
Perhaps there are other, better references for you purpose, but this is the first that comes to mind.