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## The main theorems.

Definition 3.3.9   Let A be a matrix. Denote by B the row-reduced matrix equivalent to A. The number of pivots in B is called the rank of A and is denoted by ''.

Example 3.3.10

• The restricted matrix of Examples  3.7 and  3.8 have both rank 3.
• The matrix has rank 2 (v.s. Example  2.3 and complete the computations.).

Theorem 3.3.11   A system of linear equations has a non empty set of solutions if, and only if, the rank of its augmented matrix equals the rank of its restricted matrix.

For example, a system whose matrix has an echelon form as in Figure  2 has a non empty set of solutions. A system whose matrix has an echelon form as in Figure  3 has an empty set of solutions (sometimes people say that it has no solution...); pay attention to the fact that, in the last row, there is a pivot in the last column (which belongs to the augmented matrix only). Theorem 3.3.12   Suppose that some system of linear equations has a non empty set of solutions. If there is a at least one free unknown, the set of solutions is infinite. Otherwise, the system has a unique solution.

In Fig.  4 we display the general echelon form for a system with a unique solution: As an example, v.s. Example  3.2.

In Fig.  5 we display the general echelon form for a system with an infinite set of solutions: As an example, v.s. Example  3.5.    Next: Introductory examples. Up: Solution of a system Previous: A special case: homogeneous
Noah Dana-Picard
2001-02-26