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15.7 Homogeneous Equations

The considerations above have the following implications concerning systems of homogeneous equations, in which there are no constant terms, that is, in which all right hand sides are 0.

1. If you have n variables, and n equations, and the determinant of the system is non-zero, so that the corresponding matrix is non-singular, then the origin point, or 0 vector is the only solution to the equations. It is called the trivial solution to them.

2. If there are fewer linearly independent equations than there are variables, then there are other, non-trivial solutions to the homogeneous equations. These may be found by row reduction, in parametric form, with the basis variables as parameters. Row reduction is slightly easier in this case since there is no right hand side of the equations to handle.


3. There is no way that homogeneous equations can be inconsistent; the origin is always a solution.

4. You can always add any solution of the homogeneous equations to a solution of inhomogeneous equations and the result will still be a solution of the same inhomogeneous equations.