Homogeneous Case
The homogeneous case is always consistent (because there is a trivial, all-zero solution). There are two cases, depending on the number of linearly dependent equations: either there is just the trivial solution, or there is the trivial solution plus an infinite set of other solutions.
Consider the system of linear equations: Li = 0 for 1 ≤ i ≤ M, and variables X1, X2, ..., XN, where each Li is a weighted sum of the Xis. Then X1 = X2 = ... = XN = 0 is always a solution. When M < N the system is underdetermined and there are always an infinitude of further solutions. In fact the dimension of the space of solutions is always at least N − M.
For M ≥ N, there may be no solution other than all values being 0. There will be an infinitude of other solutions only when the system of equations has enough dependencies (linearly dependent equations) that the number of independent equations is at most N − 1. But with M ≥ N the number of independent equations could be as high as N, in which case the trivial solution is the only one.
Read more about this topic: Overdetermined System
Famous quotes containing the words homogeneous and/or case:
“O my Brothers! love your Country. Our Country is our home, the home which God has given us, placing therein a numerous family which we love and are loved by, and with which we have a more intimate and quicker communion of feeling and thought than with others; a family which by its concentration upon a given spot, and by the homogeneous nature of its elements, is destined for a special kind of activity.”
—Giuseppe Mazzini (1805–1872)
“God ... created a number of possibilities in case some of his prototypes failed—that is the meaning of evolution.”
—Graham Greene (1904–1991)