Examples of applications

Water supply network example

The lesson describes the problem of a water supply network. It is shown how to know the numebr of unknowns and equations. The problem reduces to a system of linear equations, which is solved by determining the dimension of the linear subspace, which is the number of holes in the network, providing a base for this linear subspace, and a particular solution, and all this without using the algorithm. The reader will be surprised by the simplicity of the structure of all solutions.

Examples of linear systems of equations

In this lesson we describe two examples of applications. The first is a mechanical system consisting of a set of four masses connected by ropes and with two inlcined planes. The aceleration of the system and the tensions on the ropes are determined. The second example is an electric circuit that contains batteries and resistances. The circulating intensities in all holes or subcircuits are determined and the compatibility of the resulting systems of equations are analyzed.

The water supply problem

In this lesson a real water supply network problem is analysed. First, we identify the unknowns and the equations and explain their physical and engineering meaning. Later, we write system of linear equations in matrix form, discussing the importance of the node and unknown numbering in the matrix banded structure. Next, we find that the compatibility condition is a flow balance equation. Finally, we find the general solution as the sum of a linear space with dimension equal to the numnber of holes in the network, which base can be immediately determined, and finally, a particular solution is obtained, so that, there is no need of a computer to find the gneral solution of the resulting system of linear equations.