GAUSSIAN ELIMINATION
METHOD
AND
GAUSS–JORDAN ELIMINATION
METHOD
The
branch of mathematics that deals with the theory of systems of linear equations, matrices, vector spaces, determinants, and linear transformations.
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A(linear
mathematical
algebra) An
ringalgebra
and vector
over space
a field.with scalars from an associated field, the multiplication of which is of the form (a A ) (b B ) = (ab) ( AB ), where a and b are scalars and A and B are vectors.
The algebra of vectors and matrices, as distinct from the ordinary algebra of real numbers and the abstract algebra of unspecified entities.
GAUSSIAN ELIMINATION
• Echelon Forms: we solved a linear system in the unknowns x, y, and z by reducing
the augmented matrix to the form
xx00
from which the solution x=1,y=2,z=3, became evident. This is an example of a matrix that is
in reduced row-echelon form.
ECHELON FORMS:
row-echelon form and reduced row-echelon form:
To be of this form, a matrix must have the following properties:
1. If a row does not consist entirely of zeros, then the first nonzero number in the row is
a 1. We call this a leading 1.
2. If there are any rows that consist entirely of zeros, then they are grouped together at
the bottom of the matrix.
3. In any two successive rows that do not consist entirely of zeros, the leading 1 in the
lower row occurs farther to the right than the leading 1 in the higher row.
ECHELON FORMS:
• 4. Each column that contains a leading 1 has zeros everywhere else in that column.
• A matrix that has the first three properties is said to be in row-echelon form. (Thus, a
matrix in reduced row-echelon form is of necessity in row-echelon form, but not
conversely.)
ECHELON FORMS:
• EXAMPLE 1 Row-Echelon and Reduced Row-Echelon Form
The following matrices are in reduced row-echelon form.
ECHELON FORMS:
• The following matrices are in row-echelon form.
ROW-ECHELON AND REDUCED ROWECHELON FORM
• a matrix in row-echelon form has zeros below each leading 1, whereas a matrix in
reduced row-echelon form has zeros below and above each leading 1. Thus, with
any real numbers substituted for the *'s, all matrices of the following types are in
row-echelon form:
ROW-ECHELON AND REDUCED ROWECHELON FORM
• all matrices of the following types are in reduced row-echelon form:
• If, by a sequence of elementary row operations, the augmented matrix for a system of
linear equations is put in reduced row-echelon form, then the solution set of the system
will be evident by inspection or after a few simple steps.
SOLUTIONS OF LINEAR SYSTEMS
• EXAMPLE: Solutions of three Linear Systems
Suppose that the augmented matrix for a system of linear equations has been reduced by
row operations to the given reduced row-echelon form. Solve the system.
(i) Distinct Solution Case:
The corresponding system of equations is
By inspection, 𝑥1 = 5, 𝑥2 = −2, 𝑎𝑛𝑑 𝑥3 = 4
SOLUTIONS OF LINEAR SYSTEMS
• Infinite Solution Case:
The corresponding system of equations is
SOLUTIONS OF LINEAR SYSTEMS
• From this form of the equations we see that the free variable x4 can be assigned an
arbitrary value, say t, which then determines the values of the leading variables
x1,x2 and x3 . Thus there are infinitely many solutions, and the general solution is
given by the formulas
Remark The arbitrary values that are assigned to the free variables are often called
parameters. Although we shall generally use the letters r, s, t, … for the parameters,
any letters that do not conflict with the variable names may be used.
Example:
SOLUTIONS OF LINEAR SYSTEMS
• No Solution Case:
The last equation in the corresponding system of equations is
Since this equation cannot be satisfied, there is no solution to the system.
HOMOGENEOUS LINEAR SYSTEMS
• Homogeneous Linear Systems:
A system of linear equations is said to be homogeneous if the constant terms are all
zero; that is, the system has the form
Every homogeneous system of linear equations is consistent, since all such systems
have x1=0,x2=0, …, xn=0 as a solution. This solution is called the trivial solution; if
there are other solutions, they are called nontrivial solutions.
HOMOGENEOUS LINEAR SYSTEMS
• Because a homogeneous linear system always has the trivial solution, there are
only two possibilities for its solutions:
• The system has only the trivial solution.
• The system has infinitely many solutions in addition to the trivial solution.
HOMOGENEOUS LINEAR SYSTEMS
• In the special case of a homogeneous linear system of two equations in two
unknowns, say
the graphs of the equations are lines through the origin, and the trivial solution
corresponds to the point of intersection at the origin
HOMOGENEOUS LINEAR SYSTEMS
HOMOGENEOUS LINEAR SYSTEMS
Example: Solve the following homogeneous system of linear equations by using
Gauss–Jordan elimination.
The augmented matrix for the system is
HOMOGENEOUS LINEAR SYSTEMS
Reducing this matrix to reduced row-echelon form, we obtain
The corresponding system of equations is
Solving for the leading variables yields
HOMOGENEOUS LINEAR SYSTEMS
Thus, the general solution is
𝒙𝟏 = −𝒔 − 𝒕, 𝒙𝟐 = 𝒔, 𝒙𝟑 = −𝒕, 𝒙𝟒 = 𝟎, 𝒙𝟓 = 𝒕
Note that the trivial solution is obtained when 𝒔 = 𝒕 = 𝟎
HOMOGENEOUS LINEAR SYSTEMS
THEOREM 1.2.1: A homogeneous system of linear equations with more unknowns than
equations has infinitely many solutions.
THEOREM 1.2.2: FreeVariableTheorem for Homogeneous Systems
If a homogeneous linear system has n unknowns, and if the reduced row echelon form
of its augmented matrix has r nonzero rows, then the system has n − r free variables.
Note that Theorem 1.2.1 applies only to homogeneous systems—a nonhomogeneous
system with more unknowns than equations need not be consistent. However, we
will prove later that if a nonhomogeneous system with more unknowns then
equations is consistent, then it has infinitely many solutions.
HOMOGENEOUS LINEAR SYSTEMS
HINT: If you are given m homogeneous equations in n unknowns and m<n, then you
always get infinitely many solutions, since at least one variable becomes free.
For a homogeneous system, if you have m equations in n unknowns with m=n, then
you get only the trivial solution.