Systems of linear equations arise very often in engineering design and analysis. In solving for the currents or voltages in an electric circuit, loop or node equations are written which relate the currents and voltages in the circuit through the complex impedances in the circuit. If the circuit has n independent loops, there would be n loop equations which could be written, and the n loop currents would be the unknowns. Therefore, a system with n equations and n unknowns would result. The same situation would result if node equations instead of loop equations were written. In that case, the unknowns would be the node voltages. In either case, the problem of interest is to solve for the unknowns, given values for the impedances and the independent voltage or current sources.
The general form for the system of linear equations is , where X
is an n-dimensional column vector of unknowns, B is an m-dimensional
column vector of knowns, and A is an mxn matrix of known values. The
ability to solve for X depends on the relative values of m and n, and the
linear independence of the rows or columns of the A matrix. If
, then
the number of equations and the number of unknowns are equal; if the rows
(or columns) of A are linearly independent, then there is a unique solution
vector X which makes
. If
, there are more unknowns than equations.
If the rows of A are linearly independent, then there are an infinite number
of X vectors which exactly solve the problem. If
, then in general,
there is no X vector which solves the problem exactly.
In this experiment, we will explore the solution to a system of linear equations using MATLAB.