❖ The difference between \(Ax=b\) Vs \(Ax=0\) and knowing about consistent & inconsistent systems and trivial & nontrivial solutions.

\(Ax=b\) is called a Nonhomogeneous system and has a consistent or inconsistent system.

\(Ax=0\) is called a Homogeneous system and has a trivial or nontrivial solution.

❖ A linear system is called Non-homogeneous (\(Ax=b\)) if the right-hand side is a non-zero vector.

\(Ax=b\) has three possible solutions:

(1) the system has a unique (only one) solution.

(2) the system has more than one solution.

(3) the system has no solution at all.

Note:

(*) A linear system is called Consistent if there is at least one solution.

(**) A linear system is called Inconsistent if there is no solution.

❖ A linear system is called Homogeneous (\(Ax=0\)) if the right-hand side is a zero vector.

\(Ax=0\) has two possible solutions:

(1) The system has A unique solution (only one solution) called a Trivial solution.

(2) The system has infinitely many solutions (more than one solution) called Nontrivial solutions.