Ch1-LinEquations

Содержание

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Definition
An equation such as x+3y=9 is called a linear equation
(in

Ch1_ Ch1_ Definition An equation such as x+3y=9 is called a linear
two variables or unknowns).
The graph of this equation is a straight line in the xy-plane.
A pair of values of x and y that satisfy the equation is called a solution.

1.1 Matrices and Systems of Linear Equations

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Definition
A linear equation in n variables x1, x2, x3, …, xn

Ch1_ Ch1_ Definition A linear equation in n variables x1, x2, x3,
has the form a1 x1 + a2 x2 + a3 x3 + … + an xn = b
where the coefficients a1, a2, a3, …, an and b are real numbers.


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Solutions for system of linear equations

Figure 1.1
Unique solution
x + 3y =

Ch1_ Ch1_ Solutions for system of linear equations Figure 1.1 Unique solution
9
–2x + y = –4
Lines intersect at (3, 2)
Unique solution: x = 3, y = 2.

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A linear equation in three variables corresponds to a plane in three-dimensional

Ch1_ Ch1_ A linear equation in three variables corresponds to a plane
space.

※ Systems of three linear equations in three variables:

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Ch1_ Ch1_

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A solution to a system of a three linear equations will be

Ch1_ Ch1_ A solution to a system of a three linear equations
points that lie on all three planes.
The following is an example of a system of three linear equations:
How to solve a system of linear equations? For this we introduce a method called Gauss-Jordan elimination. (Section1.2)

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Definition
A matrix is a rectangular array of numbers.
The numbers in the

Ch1_ Ch1_ Definition A matrix is a rectangular array of numbers. The
array are called the elements of the matrix.

Matrices

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Submatrix

Row and Column

Ch1_ Ch1_ Submatrix Row and Column

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matrix of coefficients and augmented matrix

Relations between system of linear equations

Ch1_ Ch1_ matrix of coefficients and augmented matrix Relations between system of linear equations and matrices
and matrices


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Elementary Transformation
Interchange two equations.
Multiply both sides of an equation by a

Ch1_ Ch1_ Elementary Transformation Interchange two equations. Multiply both sides of an
nonzero constant.
Add a multiple of one equation to another equation.


Elementary Row Operation
Interchange two rows of a matrix.
Multiply the elements of a row by a nonzero constant.
Add a multiple of the elements of one row to the corresponding elements of another row.

Elementary Row Operations of Matrices

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Example 1

Solving the following system of linear equation.

Ch1_ Ch1_ Example 1 Solving the following system of linear equation.

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Example 2

Solving the following system of linear equation.

Solution

Ch1_ Ch1_ Example 2 Solving the following system of linear equation. Solution

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Example 3

Solve the system

Solution

Ch1_ Ch1_ Example 3 Solve the system Solution

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Summary

Use row operations to [A: B] :

Def. [In : X] is called

Ch1_ Ch1_ Summary Use row operations to [A: B] : Def. [In
the reduced echelon form of [A : B].

Note. 1. If A is the matrix of coefficients of a system of n equations in n variables that has a unique solution, then A is row equivalent to In (A ≈ In).
2. If A ≈ In, then the system has unique solution.

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Example 4 Many Systems

Solving the following three systems of linear equation, all

Ch1_ Ch1_ Example 4 Many Systems Solving the following three systems of
of which have the same matrix of coefficients.

Solution

The solutions to the three systems are

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Homework

Exercises will be given by the teachers of the practical classes.

Ch1_ Ch1_ Homework Exercises will be given by the teachers of the practical classes.

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1.2 Gauss-Jordan Elimination

Definition
A matrix is in reduced echelon form if
Any rows consisting

Ch1_ Ch1_ 1.2 Gauss-Jordan Elimination Definition A matrix is in reduced echelon
entirely of zeros are grouped at the bottom of the matrix.
The first nonzero element of each other row is 1. This element is called a leading 1.
The leading 1 of each row after the first is positioned to the right of the leading 1 of the previous row.
All other elements in a column that contains a leading 1 are zero.

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Examples for reduced echelon form

(✓)

(✓)

(?)

(?)

elementary row operations,reduced echelon form
The reduced echelon form

Ch1_ Ch1_ Examples for reduced echelon form (✓) (✓) (?) (?) elementary
of a matrix is unique.

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Gauss-Jordan Elimination

System of linear equations ⇒ augmented matrix ⇒ reduced echelon form ⇒

Ch1_ Ch1_ Gauss-Jordan Elimination System of linear equations ⇒ augmented matrix ⇒
solution

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Example 1

Use the method of Gauss-Jordan elimination to find reduced echelon form

Ch1_ Ch1_ Example 1 Use the method of Gauss-Jordan elimination to find
of the following matrix.

Solution

The matrix is the reduced echelon form of the given matrix.

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Example 2

Solve, if possible, the system of equations

Solution

The general solution to the

Ch1_ Ch1_ Example 2 Solve, if possible, the system of equations Solution
system is

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Example 3

Solve the system of equations

Solution

⇒ many sol.

Ch1_ Ch1_ Example 3 Solve the system of equations Solution ⇒ many sol.

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Example 4

Solve the system of equations

Solution

Ch1_ Ch1_ Example 4 Solve the system of equations Solution

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Example 5

This example illustrates a system that has no solution. Let us

Ch1_ Ch1_ Example 5 This example illustrates a system that has no
try to solve the system

Solution

The system has no solution.

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Homogeneous System of linear Equations

Definition
A system of linear equations is said

Ch1_ Ch1_ Homogeneous System of linear Equations Definition A system of linear
to be homogeneous if all the constant terms are zeros.

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Homogeneous System of linear Equations

Note. Non trivial solution

Ch1_ Ch1_ Homogeneous System of linear Equations Note. Non trivial solution

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Homework

Exercise will be given by the teachers of the practical classes.

Ch1_ Ch1_ Homework Exercise will be given by the teachers of the practical classes.

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1.3 Gaussian Elimination

Definition
A matrix is in echelon form if
Any rows consisting entirely

Ch1_ 1.3 Gaussian Elimination Definition A matrix is in echelon form if
of zeros are grouped at the bottom of the matrix.
The first nonzero element of each row is 1. This element is called a leading 1.
The leading 1 of each row after the first is positioned to the right of the leading 1 of the previous row.
(This implies that all the elements below a leading 1 are zero.)

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Example 6

Solving the following system of linear equations using the

Ch1_ Example 6 Solving the following system of linear equations using the
method of Gaussian elimination.

Solution

We have arrived at the echelon form.

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The corresponding system of equation is

Ch1_ The corresponding system of equation is

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Example 7

Solving the following system of linear equations using the

Ch1_ Example 7 Solving the following system of linear equations using the
method of Gaussian elimination, performing back substitution using matrices.

Solution

This marks the end of the forward elimination of variables from equations. We now commence the back substitution using matrices.

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