next up previous contents
Next: Introductory examples. Up: Solution of a system Previous: A special case: homogeneous

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 ``$\rank A$''.

Example 3.3.10       

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.
\begin{figure}
\mbox{\epsfig{file=pivots1.eps,height=5cm} }
\end{figure}

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.
\begin{figure}
\mbox{\epsfig{file=pivots2.eps,height=5cm} }
\end{figure}

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.
\begin{figure}
\mbox{\epsfig{file=pivots3.eps,height=4cm} }
\end{figure}

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.
\begin{figure}
\mbox{\epsfig{file=pivots4.eps,height=5cm} }
\end{figure}

As an example, v.s. Example  3.5.


next up previous contents
Next: Introductory examples. Up: Solution of a system Previous: A special case: homogeneous
Noah Dana-Picard
2001-02-26