next up previous contents
Next: The Gauss-Jordan algorithm. Up: Systems of linear equations. Previous: Systems of linear equations.

Systems of equations and associated matrices.

A linear equation in n unknowns $x_1, \dots, x_n$ is an equation of the form

\begin{displaymath}a_1x_1+a_2x_2+ \dots +a_nx_n=b
\end{displaymath}

where the coefficients $a_1,a_2, \dots, a_n$ and b are given numbers (i.e. elements of some field $\mathbb{K} $; in this tutorial, this field will generally be $\mathbb{R} $ or $\mathbb{C} $).

When the number of unknowns is little, the unknowns can be denoted by $x,y,z, \dots$.

A system of linear equations is a set of linear equations with the same unknowns and coefficients in the same field:

 \begin{displaymath}
\begin{cases}
a_{11}x_1+ a_{12}x_2+ a_{13}x_3+ \dots + a_{1n...
...}x_1+ a_{p2}x_2+ a_{p3}x_3+ \dots + a_{pn}x_n =b_p
\end{cases}\end{displaymath} (3.1)

A solution of the given system is an n-tuple $(\xi_1, \xi_2, \dots, \xi_n)$ of elements of $\mathbb{K} $ such that, when replacing each xi by the corresponding $\xi_i$ we transform the equations into identities.

Example 3.1.1   Consider the following system of linear equations:

 \begin{displaymath}
\begin{cases}
x_1+2x_2-3x_3= 23 \\ 3x_1-2x_2+x_3=-11 \\ -x_1-2x_2+x_3=-15
\end{cases}\end{displaymath} (3.2)

The triple (1,5,-4) is a solution of the system ( 2), and the triple (1,2,1) is not a solution of the system ( 2). To check this, just replace into the equations.

To this system we associate the augmented matrix of the system:


 \begin{displaymath}
\begin{pmatrix}a_{11} & a_{12} &a_{13} & \dots & a_{1n} & b_...
... a_{p1} & a_{p2} &a_{p3} & \dots & a_{pn} & b_p
\end{pmatrix}\end{displaymath} (3.3)

Erasing the last column of the augmented matrix, we get the restricted matrix or coefficient matrix of the system:


 \begin{displaymath}
\begin{pmatrix}a_{11} & a_{12} &a_{13} & \dots & a_{1n} \\
...
...s \\
a_{p1} & a_{p2} &a_{p3} & \dots & a_{pn}
\end{pmatrix}\end{displaymath} (3.4)

Example 3.1.2   The augmented matrix of the system ( 2) is

\begin{displaymath}\begin{pmatrix}1& 2 &-3 & 23 \\ 3 & -2 & 1 & -11 \\ -1 & -2 & 1 & -15 \end{pmatrix}\end{displaymath}

and the restricted matrix of the system ( 2) is

\begin{displaymath}\begin{pmatrix}1& 2 &-3 \\ 3 & -2 & 1 \\ -1 & -2 & 1 \end{pmatrix}\end{displaymath}

In order to distinguish the restricted part of the augmented matrix from the last column, which has a different meaning, we introduce sometimes a separation line into the matrix, as displayed in the following matrix:


  
Figure 1: The augmented matrix of a system
\begin{figure}
\mbox{\epsfig{file=AugmentedMatrix.eps,height=4cm} }
\end{figure}

Definition 3.1.3   A matrix is in row-echelon form if it fulfills the following requirements:
1.
If there are rows containing only zeros, they are at the bottom of the matrix;
2.
If two successive rows contain non zero entries, the second row starts with more zeros than the first one.

Example 3.1.4       

$\begin{pmatrix}1& 1 & 2 & 3 \\ 0 & 2 & -1 & 1 \\ 0 & 0 & 0 & 2 \end{pmatrix}$ is in row-echelon form.

$\begin{pmatrix}1& 1 & 2 & 3 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & -1 & 2 \end{pmatrix}$ is not in row-echelon form.

Definition 3.1.5   A matrix is in row-reduced echelon form if it fulfills the following requirements:
1.
The matrix is in row-echelon form;
2.
The leftmost non zero entry (called leading entry) on each row is equal to 1;
3.
If a column contains a leading entry, then all other entrues in this column are equal to 0.

Example 3.1.6       

$\begin{pmatrix}1& 1 & 2 & 3 \\ 0 & 2 & -1 & 1 \\ 0 & 0 & 0 & 2 \end{pmatrix}$ is in row-echelon form, but is not row-reduced.

$\begin{pmatrix}1& 1 & 0 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{pmatrix}$ is not in row-reduced echelon form, as in the entries above the pivot in the last column are not equal to 0.


next up previous contents
Next: The Gauss-Jordan algorithm. Up: Systems of linear equations. Previous: Systems of linear equations.
Noah Dana-Picard
2001-02-26