Linear systems of equations

Homogeneous linear systems of equations

In this lesson we show how the orthogonal algorithm can be used to solve a linear system of homogeneous equations. It is shown that the solution of a linear system of homogeneous equations is the orthogonal subspace to the linear subspace generated by the row vectors of the coefficients of the system of equations. Thus, a direct application of the algorithm leads to the solution of this problem.

Solving complete linear systems of equations

In this lesson we explain how to use the orthogonalization algorithm to solve a complete linear system of equations. First, a virtual unknown is used to transform the system into an homogeneous system plus an additional equation. Later the homogeneous system is solved, and finally, the last equation is imposed. This permits obtaining all solutions as the sum of a linear space plus a particular solution.

Compatibility of a linear system of equations

In this lesson we analize the compatibility of a linear system of equations without solving the problem. It is shown that the shystem is compatible if the column vector of independent terms belongs to the linear subspace generated by the columns of the linear system of equations. Thus, the compatibility problem is reduced to a mwmbership problem.