Ax=b Vs Ax=0, consistent & inconsistent, trivial & nontrivial
❖ 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.
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• Linear Algebra
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