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Definition 3.3.9
Let
A be a matrix. Denote by
B the rowreduced 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.
Figure 2:
Same rank.

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).
Figure 3:
Different ranks.

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:
Figure 4:
A matrix for a finite set of solutions.

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:
Figure 5:
A matrix for 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 DanaPicard
20010226