Download Chap 1.I.1 - Gauss`s Method

Chap 1.I.1 - Gauss’s Method
MTH 214 - Linear Algebra
Dr. Lew
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We’ll start with basic definitions and results regarding linear equations.
Linear Equation
A linear equation in the variables x1 , x2 , . . . , xn is an equation
a1 x1 + a2 x2 + · · · + an xn = b
where a1 , a2 , . . . , an are the coefficients and b is called the constant.
Example
4x2 + 5x3 + 2 = x1 is a linear equation since it can be written
−x1 + 4x2 + 5x3 = −2 or x1 − 4x2 − 5x3 = 2.
4x1 − 5x2 = x1 x3 is a nonlinear equation, as is the equation
√
x2 = 2 x1 + 3
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System of Linear Equations
A system of linear equations (or linear system) is a collection
of one or more linear equations involving the same set of variables
x1 , x2 , . . . , xn .
Example
A system of three linear equations in four variables is given below:
2x1 − 3x2 + x3 − 2x4 = 8
− x1 + 2x2 − 4x3
= −5
3x1 − 2x2 − x3 + 3x4 = 2
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Solution
A solution to a linear system with variables x1 , x2 , . . . , xn is a set of
values (s1 , s2 , . . . , sn ) such that setting xi = si for i = 1, 2, . . . , n in
each linear equation results in a true statement.
Solution Set
The set of all possible solutions to a linear system is called the solution
set of the linear system.
Equivalent Systems
Two linear systems are called equivalent if they have the same solution
set.
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Example 1
x1 + x2 = 10
−x1 + x2 = 0
This linear system has EXACTLY ONE solution of (5, 5).
Example 2
x1 − 2x2 = −3
2x1 − 4x2 = 8
This linear system has NO SOLUTIONS. There are no values for x1
and x2 that satisfy both equations simultaneously.
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Example 3
x1 + x2 = 3
−2x1 − 2x2 = −6
This linear system has INFINITELY MANY solutions. Any solution
will have the form (s, 3 − s) where s is any real number.
FACT
The solution set of a linear system will contain either:
no solution, or
exactly one solution, or
infinitely many solutions.
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Consistent Systems
A linear system is called consistent if it has either one solution or infinitely many solutions.
A linear system with no solutions is called inconsistent.
Solving a Linear System. The strategy for solving a linear system is to
replace it with an equivalent system (one with the same solution set) that
is easier to solve.
We can do this using three algebraic operations applied to the linear
system. We can illustrate them by an example.
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Suppose we want to determine the solution set for the linear system
x1 − 2x2 + x3 = 0
2x2 − 8x3 = 8
−4x1 + 5x2 + 9x3 = −9
When one of these three operations is applied to a linear system, the
altered linear system will be equivalent to the original one.
The goal is to apply these operations in a such a manner that we alter the
linear system into a simpler one whose solutions can be easily found.
1. First, multiply the first equation by 4 and add it to the third
equation:
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x1 − 2x2 + x3 = 0
2x2 − 8x3 = 8
− 3x2 + 13x3 = −9
2. Next, multiply the second equation by 1/2:
x1 − 2x2 + x3 = 0
x2 − 4x3 = 4
− 3x2 + 13x3 = −9
3. Multiply the second equation by 3 and add it to the third
equation:
x1 − 2x2 + x3 = 0
x2 − 4x3 = 4
x3 = 3
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4. Now multiply the third equation by 4 and add it to the second
equation, and multiply the third equation by −1 and add it to
the first equation:
x1 − 2x2
x2
= −3
= 16
x3 = 3
5. Finally, multiply the second equation by 2 and add it to the first
equation:
x1
= 29
x2
= 16
x3 = 3
So there is exactly one solution to the original linear system.
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In fact, this solution is also the solution to each intermediate linear
system created along the way. In other words, all of the linear systems
created are equivalent to each other. This is an example of the following
fundamental result.
Theorem
If a linear system is changed to another via one of these operations:
Two equations are interchanged with each other.
An equation is multiplied through by a nonzero constant.
An equation is replaced by the sum of itself and a multiple of another.
then the two linear systems have the same solution set.
The justification for this result appears in the book.
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The three operations stated above are called elementary reduction
operations, or elementary row operations, or Gaussian operations.
Interchange: Interchange the positions of two equations. The
interchange of the ith equation and the jth equation is written as
ρi ↔ ρj .
Replacement: Replace one equation by the sum of itself and a multiple
of another equation. If the ithe equation is multiplied by a constant C
and added to the jth equation, then denote this operation as
Cρi + ρj .
Scaling: Multiply an equation through by a nonzero constant. If the ith
equation is multiplied by a constant C, then denote this operation as
Cρi .
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The process of solving a linear system by applying elementary reduction
operations is called Gauss’s Method, or Gauss Reduction.
We can summarize Gauss’s Method applied to our previous example
using the following notation:
x1 − 2x2 + x3 = 0
2x2 − 8x3 = 8
−4x1 + 5x2 + 9x3 = −9
−→
x1 − 2x2 + x3 = 0
x2 − 4x3 = 4
− 3x2 + 13x3 = −9
1
ρ
2 2
−→
x1 − 2x2 + x3 = 0
2x2 − 8x3 = 8
− 3x2 + 13x3 = −9
4ρ1 +ρ3
3ρ2 +ρ3
−→
x1 − 2x2 + x3 = 0
x2 − 4x3 = 4
x3 = 3
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x1 − 2x2 + x3 = 0
x2 − 4x3 = 4
x3 = 3
2ρ2 +ρ1
−→
x1 − 2x2
x2
4ρ3 +ρ2
−→
−1ρ3 +ρ1
x1
x2
= −3
= 16
x3 = 3
= 29
= 16
x3 = 3
Example
Applying Gauss reduction, verify we get the following equivalent linear
systems:
−x − y + 3z = 3
x
+ z= 3
3x − y + 7z = 15
−→ · · · −→
−x − y + 3z = 3
− y + 4z = 6
0=0
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−x − y + 3z = 3
− y + 4z = 6
0=0
This linear system has infinitely many solutions. This follows from observing that once a value for z is chosen, then the second equation determines the value of y.
Once z and y are determined, they can be used in the first equation to
determine the value of x.
For example, setting z = 0 gives y = −6 and x = 3. So a solution is
(3, −6, 0).
If we set z = 1, then this will give the solution (2, −2, 1).
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We will generally (but not always) encounter infinitely many solutions for
a linear system when there are more variables than equations.
Gauss’s Method uses elementary reduction operations to get a linear
system where back-substitution can be used to find the solutions(s) to the
original linear system. The form of the linear system we obtain is given a
particular name.
Echelon Form
In a row of a linear system, the first variable with a nonzero coefficient
is called the row’s leading variable.
A system is in echelon form if each leading variable in a linear equation
is to the right of the leading variable in any linear equation above it and
any “all-zero” equations appear at the bottom.
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For example, the linear system below is in echelon form:
−x − y + 3z = 3
− y + 4z = 6
0=0
while this linear system is not:
4y + z = 20
2x − 2y + z = 0
x
+ z= 5
y + 2z = 7
(END)
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