Homogenous Systems

A homogenous systems is just a system of linear equations where all constants on the right side of the equals sign are zero. In other words, the linear system of the form Ax = 0.

A homogeneous system always has the solution x = 0. This is called the trivial solution. Any nonzero solution is called nontrivial.

If a homogenous system has a nontrivial solution, this is equivalent to saying:

• There is a free variable
• A has a column without a pivot.

Finding the solutions to a homogenous system is effectively finding the set of vectors that get ‘squished’ to a lower dimension by the linear transformation represented by a certain matrix A.

The set of solutions found by solving a homogenous system is a span.