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# Systems of equations and associated matrices.

A linear equation in n unknowns is an equation of the form where the coefficients and b are given numbers (i.e. elements of some field ; in this tutorial, this field will generally be or ).

When the number of unknowns is little, the unknowns can be denoted by .

A system of linear equations is a set of linear equations with the same unknowns and coefficients in the same field: (3.1)

A solution of the given system is an n-tuple of elements of such that, when replacing each xi by the corresponding we transform the equations into identities.

Example 3.1.1   Consider the following system of linear equations: (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: (3.3)

Erasing the last column of the augmented matrix, we get the restricted matrix or coefficient matrix of the system: (3.4)

Example 3.1.2   The augmented matrix of the system ( 2) is and the restricted matrix of the system ( 2) is 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: 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 is in row-echelon form. 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 is in row-echelon form, but is not row-reduced. is not in row-reduced echelon form, as in the entries above the pivot in the last column are not equal to 0.    Next: The Gauss-Jordan algorithm. Up: Systems of linear equations. Previous: Systems of linear equations.
Noah Dana-Picard
2001-02-26