Fall 2017

Systems of LinearEquations

Authors:Alexander Knop

Institute:UC San Diego

Grading Policy

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

▶ Homework 5 to 10% of the course grade.

▶ Quizzes 5 to 10% of the course grade.▶ MatLab 5 to 10% of the course grade.▶ Two midterm exams 20% of the course grade each.▶ The final exam 35% of the course grade.

ORGANIZATION OF THE COURSE | Alexander Knop 2

Grading Policy

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

▶ Homework 5 to 10% of the course grade.▶ Quizzes 5 to 10% of the course grade.

▶ MatLab 5 to 10% of the course grade.▶ Two midterm exams 20% of the course grade each.▶ The final exam 35% of the course grade.

ORGANIZATION OF THE COURSE | Alexander Knop 2

Grading Policy

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

▶ Homework 5 to 10% of the course grade.▶ Quizzes 5 to 10% of the course grade.▶ MatLab 5 to 10% of the course grade.

▶ Two midterm exams 20% of the course grade each.▶ The final exam 35% of the course grade.

ORGANIZATION OF THE COURSE | Alexander Knop 2

Grading Policy

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

▶ Homework 5 to 10% of the course grade.▶ Quizzes 5 to 10% of the course grade.▶ MatLab 5 to 10% of the course grade.▶ Two midterm exams 20% of the course grade each.

▶ The final exam 35% of the course grade.

ORGANIZATION OF THE COURSE | Alexander Knop 2

Grading Policy

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

▶ Homework 5 to 10% of the course grade.▶ Quizzes 5 to 10% of the course grade.▶ MatLab 5 to 10% of the course grade.▶ Two midterm exams 20% of the course grade each.▶ The final exam 35% of the course grade.

ORGANIZATION OF THE COURSE | Alexander Knop 2

Homework

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

Every Friday (except the first one) you will have a hometask thatcover material from previous Friday, Monday, and Wednesday.

We will use mymathlab.com for them.You will need an access code for this.

ORGANIZATION OF THE COURSE | Alexander Knop 3

Homework

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

Every Friday (except the first one) you will have a hometask thatcover material from previous Friday, Monday, and Wednesday.

We will use mymathlab.com for them.

You will need an access code for this.

ORGANIZATION OF THE COURSE | Alexander Knop 3

Homework

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

Every Friday (except the first one) you will have a hometask thatcover material from previous Friday, Monday, and Wednesday.

We will use mymathlab.com for them.You will need an access code for this.

ORGANIZATION OF THE COURSE | Alexander Knop 3

Quizzes

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

Every Friday (except the first one) you will have a quiz that covermaterial from previous Friday, Monday, and Wednesday.

Quiz will take place on the last ten minutes of the class.For grading these quizzed I will use Gradescope.

ORGANIZATION OF THE COURSE | Alexander Knop 4

Quizzes

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

Every Friday (except the first one) you will have a quiz that covermaterial from previous Friday, Monday, and Wednesday.Quiz will take place on the last ten minutes of the class.

For grading these quizzed I will use Gradescope.

ORGANIZATION OF THE COURSE | Alexander Knop 4

Quizzes

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

Every Friday (except the first one) you will have a quiz that covermaterial from previous Friday, Monday, and Wednesday.Quiz will take place on the last ten minutes of the class.For grading these quizzed I will use Gradescope.

ORGANIZATION OF THE COURSE | Alexander Knop 4

Discussions and Questions

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

▶ email: aknop@ucsd.edu

▶ Piazza site:https://piazza.com/uc_san_diego/fall2017/math18/home

ORGANIZATION OF THE COURSE | Alexander Knop 5

Discussions and Questions

All this information is available on the website:math.ucsd.edu/∼aknop/ucsd/18/2017/

▶ email: aknop@ucsd.edu▶ Piazza site:

https://piazza.com/uc_san_diego/fall2017/math18/home

ORGANIZATION OF THE COURSE | Alexander Knop 5

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

1 Is 2x1 + 4x3 + x5 = x6 a linear equation in the variables x1, …, x6?

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

1 2x1 + 4x3 + x5 = x6 is a linear equation in the variables x1, …, x6 since itis equivalent to 2x1 + 0x2 + 4x3 + x5 + (−1)x6 = 0;

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

1 2x1 + 4x3 + x5 = x6 is a linear equation in the variables x1, …, x6 since itis equivalent to 2x1 + 0x2 + 4x3 + x5 + (−1)x6 = 0;

2 Is x1 +√5x2 =

√6(1− x3) a linear equation in the variables x1, x2, x3.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

1 2x1 + 4x3 + x5 = x6 is a linear equation in the variables x1, …, x6 since itis equivalent to 2x1 + 0x2 + 4x3 + x5 + (−1)x6 = 0;

2 x1 +√5x2 =

√6(1− x3) is a linear equation in variables x1, x2, x3 since

it is equivalent to x1 +√5x2 +

√6x3 =

√6.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

1 Is x1 = x2x3 + x4 a linear equation in the variables x1, …, x4?

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

1 x1 = x2x3 + x4 is not a linear equation in the variables x1, …, x4 since wehave a summand x2x3;

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

1 x1 = x2x3 + x4 is not a linear equation in the variables x1, …, x4 since wehave a summand x2x3;

2 Is x1 =√x2 a linear equation in the variables x1 and x2?

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

1 x1 = x2x3 + x4 is not a linear equation in the variables x1, …, x4 since wehave a summand x2x3;

2 x1 =√x2 is not a linear equation in the variables x1 and x2 because of

presence of √x2.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

1 Is x1 = x2x3 + x4 a linear equation in the variables x1, x2?

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

1 x1 = x2x3 + x4 is a linear equation in the variables x1, x2;

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

1 x1 = x2x3 + x4 is a linear equation in the variables x1, x2;2 Is x + y = 3 a linear equation in the variables x, y?

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Linear Equations

DEFINITION

A linear equation in the variables x1, …, xn is an equation that can be writtenin the form

a1x1 + a2x2 + · · ·+ anxn = bwhere a1, …, an, and b are real numbers (i.e. a1, . . . , an, b ∈ R).

EXAMPLE

1 x1 = x2x3 + x4 is a linear equation in the variables x1, x2;2 x + y = 3 is a linear equation in the variables x, y.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 6

Systems of Linear Equations

DEFINITION

A system of linear equations in the variables x1, …, xn is a collection of one ormore linear equations in the variables x1, …, xn.

EXAMPLE

2x1 − x2 + 3x3 = 8

x1 − 4x3 = −7

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 7

Systems of Linear Equations

DEFINITION

A system of linear equations in the variables x1, …, xn is a collection of one ormore linear equations in the variables x1, …, xn.

EXAMPLE

2x1 − x2 + 3x3 = 8

x1 − 4x3 = −7

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 7

Solutions of a System of Linear Equations

DEFINITION

A solution of of the system of linear equations in the variables x1, …, xn is alist (s1, . . . , sn) of real numbers such that each equation in the system becametrue if we substitute the values s1, …, sn to the variables x1, …, xn respectively.

EXAMPLE

(5, 6.5, 3) is a solution of

2x1 − x2 + 1.5x3 = 8

x1 − 4x3 = −7

since 2 · 5− 6.5 + 1.5 · 3 = 8 and 5− 4 · 3 = −7.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 8

Solutions of a System of Linear Equations

DEFINITION

A solution of of the system of linear equations in the variables x1, …, xn is alist (s1, . . . , sn) of real numbers such that each equation in the system becametrue if we substitute the values s1, …, sn to the variables x1, …, xn respectively.

EXAMPLE

(5, 6.5, 3) is a solution of

2x1 − x2 + 1.5x3 = 8

x1 − 4x3 = −7

since 2 · 5− 6.5 + 1.5 · 3 = 8 and 5− 4 · 3 = −7.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 8

Solutions sets

DEFINITION

The set of all possible solutions of a system of linear equations is called asolution set.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 9

Solutions of a System of Linear EquationsSolution set of the system

x1 − 2x2 = −1

−x1 + 3x2 = 3

is {(2, 3)}.

0

1

2

3

0 1 2 3 4 5

x1

x2

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 10

Solutions of a System of Linear EquationsSolution set of the system

x1 − 2x2 = −1

x1 − 2x2 = 0

is empty.

0

1

2

3

0 1 2 3 4 5

x1

x2

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 11

Solutions of a System of Linear EquationsSolution set of the system

x1 − 2x2 = −1

2x1 − 4x2 = −2

is equal to {(−1 + 2x2, x2) | x2 ∈ R}.

0

1

2

3

0 1 2 3 4 5

x1

x2

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 12

Solutions of a System of Linear Equations

A system of two linear equations in two variables has1 no solution, or2 exactly one solution, or3 infinitely many solutions.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 13

Solutions of a System of Linear Equations

A system of linear equations has1 no solution, or2 exactly one solution, or3 infinitely many solutions.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 13

Matrix notation

The most important information about a system of equations wecan write in a rectangular array.

EXAMPLE

The systemx1 − 2x2 + 4x3 = −1

2x1 + x3 = −2

We may rewrite as [1 −2 4 −12 0 1 −2

]

We call such a matrix augmented matrix and this matrix withoutlast column is called coefficient matrix.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 14

Solving a Linear System

Let us solve the following system.

x1 − 2x2 + x3 = 0

2x2 − 8x3 = 8

−4x1 + 5x2 + 9x3 = −9

1 −2 1 00 2 −8 8−4 5 9 −9

Add 4 times the first equation to the third equation.

4 · [equation 1:]+[equation 3:]

[new equation 3:]

4x1 − 8x2 + 4x3 = 0−4x1 + 5x2 + 9x3 = −9

−3x2 + 13x3 = −9

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

Solving a Linear System

Let us solve the following system.

x1 − 2x2 + x3 = 0

2x2 − 8x3 = 8

−4x1 + 5x2 + 9x3 = −9

1 −2 1 00 2 −8 8−4 5 9 −9

Add 4 times the first equation to the third equation.

4 · [equation 1:]+[equation 3:]

[new equation 3:]

4x1 − 8x2 + 4x3 = 0−4x1 + 5x2 + 9x3 = −9

−3x2 + 13x3 = −9

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

Solving a Linear System

Let us solve the following system.

x1 − 2x2 + x3 = 0

2x2 − 8x3 = 8

− 3x2 + 13x3 = −9

1 −2 1 00 2 −8 80 −3 13 −9

Divide equation 2 by 2.

x1 − 2x2 + x3 = 0

x2 − 4x3 = 4

− 3x2 + 13x3 = −9

1 −2 1 00 1 −4 40 −3 13 −9

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

Solving a Linear System

Let us solve the following system.

x1 − 2x2 + x3 = 0

2x2 − 8x3 = 8

− 3x2 + 13x3 = −9

1 −2 1 00 2 −8 80 −3 13 −9

Divide equation 2 by 2.

x1 − 2x2 + x3 = 0

x2 − 4x3 = 4

− 3x2 + 13x3 = −9

1 −2 1 00 1 −4 40 −3 13 −9

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

Solving a Linear System

Let us solve the following system.

x1 − 2x2 + x3 = 0

x2 − 4x3 = 4

− 3x2 + 13x3 = −9

1 −2 1 00 1 −4 40 −3 13 −9

Add 3 times equation 2 to equation 3.

3 · [equation 2:]+[equation 3:]

[new equation 3:]

3x2 − 12x3 = 12−3x2 + 13x3 = −9

x3 = 3

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

Solving a Linear System

Let us solve the following system.

x1 − 2x2 + x3 = 0

x2 − 4x3 = 4

− 3x2 + 13x3 = −9

1 −2 1 00 1 −4 40 −3 13 −9

Add 3 times equation 2 to equation 3.

3 · [equation 2:]+[equation 3:]

[new equation 3:]

3x2 − 12x3 = 12−3x2 + 13x3 = −9

x3 = 3

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

Solving a Linear System

Let us solve the following system.

x1 − 2x2 + x3 = 0

x2 − 4x3 = 4

x3 = 3

1 −2 1 00 1 −4 40 0 1 3

Add 2 times equation 2 to equation 1.

2 · [equation 2:]+[equation 1:]

[new equation 1:]

2x2 − 8x3 = 8x1 −2x2 + x3 = 0

x1 −7x3 = 8

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

Solving a Linear System

Let us solve the following system.

x1 − 2x2 + x3 = 0

x2 − 4x3 = 4

x3 = 3

1 −2 1 00 1 −4 40 0 1 3

Add 2 times equation 2 to equation 1.

2 · [equation 2:]+[equation 1:]

[new equation 1:]

2x2 − 8x3 = 8x1 −2x2 + x3 = 0

x1 −7x3 = 8

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

Solving a Linear System

Let us solve the following system.

x1 − 7x3 = 8

x2 − 4x3 = 4

x3 = 3

1 0 −7 80 1 −4 40 0 1 3

Add 4 times equation 3 to equation 2.

4 · [equation 3:]+[equation 1:]

[new equation 2:]

4x3 = 12x2 − 3x3 = 4

x2 = 16

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

Solving a Linear System

Let us solve the following system.

x1 − 7x3 = 8

x2 − 4x3 = 4

x3 = 3

1 0 −7 80 1 −4 40 0 1 3

Add 4 times equation 3 to equation 2.

4 · [equation 3:]+[equation 1:]

[new equation 2:]

4x3 = 12x2 − 3x3 = 4

x2 = 16

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

Solving a Linear System

Let us solve the following system.

x1 − 7x3 = 8

x2 = 16

x3 = 3

1 0 −7 80 1 0 160 0 1 3

Add 7 times equation 3 to equation 1.

7 · [equation 3:]+[equation 1:]

[new equation 1:]

7x3 = 21x1 − 7x3 = 8

x1 = 29

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

Solving a Linear System

Let us solve the following system.

x1 − 7x3 = 8

x2 = 16

x3 = 3

1 0 −7 80 1 0 160 0 1 3

Add 7 times equation 3 to equation 1.

7 · [equation 3:]+[equation 1:]

[new equation 1:]

7x3 = 21x1 − 7x3 = 8

x1 = 29

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

Solving a Linear System

Let us solve the following system.

x1 = 29

x2 = 16

x3 = 3

1 0 0 290 1 0 160 0 1 3

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 15

General Case

ELEMENTARY ROW OPERATIONS

Replacement Replace one row by the sum of itself and multiple of anotherrow.

Interchange Interchange two rows.Scaling Multiply a row by a nonzero constant.

Note that we may apply these operations to any matrix. Not onlyto the augmented matrix of a system of equations.

DEFINITION

We call two matrices row equivalent if there is a sequence of elementary rowoperations that transform one matrix into the other.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 16

General Case

ELEMENTARY ROW OPERATIONS

Replacement Replace one row by the sum of itself and multiple of anotherrow.

Interchange Interchange two rows.Scaling Multiply a row by a nonzero constant.

Note that we may apply these operations to any matrix. Not onlyto the augmented matrix of a system of equations.

DEFINITION

We call two matrices row equivalent if there is a sequence of elementary rowoperations that transform one matrix into the other.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 16

General Case

ELEMENTARY ROW OPERATIONS

Replacement Replace one row by the sum of itself and multiple of anotherrow.

Interchange Interchange two rows.Scaling Multiply a row by a nonzero constant.

Note that we may apply these operations to any matrix. Not onlyto the augmented matrix of a system of equations.

DEFINITION

We call two matrices row equivalent if there is a sequence of elementary rowoperations that transform one matrix into the other.

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 16

Conservation of a Solutions Set

THEOREM

If the augmented matrices of two systems are row equivalent, then two systemshave the same solutions sets.

TWO FUNDAMENTAL QUESTIONS ABOUT A LINEAR SYSTEM

1 Is the system consistent; that is, does at least one solution exist?2 If solution exists, is it the only one?; that is, is the solution unique?

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 17

Conservation of a Solutions Set

THEOREM

If the augmented matrices of two systems are row equivalent, then two systemshave the same solutions sets.

TWO FUNDAMENTAL QUESTIONS ABOUT A LINEAR SYSTEM

1 Is the system consistent; that is, does at least one solution exist?2 If solution exists, is it the only one?; that is, is the solution unique?

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 17

The Fundamental Questions

EXAMPLE

Let us answer this questions about the system

x2 − 4x3 = 8

2x1 − 3x2 + 2x3 = 1

5x1 − 8x2 + 7x3 = 1

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 18

The Fundamental Questions

EXAMPLE

Let us answer this questions about the system

x2 − 4x3 = 8

2x1 − 3x2 + 2x3 = 1

5x1 − 8x2 + 7x3 = 1

Augmented matrix of this system is0 1 −4 82 −3 2 15 −8 7 1

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 18

The Fundamental Questions

EXAMPLE

Let us answer this questions about the system0 1 −4 82 −3 2 15 −8 7 1

Let us interchange rows 1 and 2.2 −3 2 1

0 1 −4 85 −8 7 1

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 18

The Fundamental Questions

EXAMPLE

Let us answer this questions about the system2 −3 2 10 1 −4 85 −8 7 1

Add −5/2 times row 1 to row 3.2 −3 2 1

0 1 −4 80 −1/2 2 −3/2

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 18

The Fundamental Questions

EXAMPLE

Let us answer this questions about the system2 −3 2 10 1 −4 80 −1/2 2 −3/2

Add 1/2 times row 2 to row 3.2 −3 2 1

0 1 −4 80 0 0 5/2

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 18

The Fundamental Questions

EXAMPLE

Let us answer this questions about the system2 −3 2 10 1 −4 80 0 0 5/2

The corresponding system is

2x1 − 3x2 + 2x3 = 1

x2 − 4x3 = 8

+ 0 = 5/2

SYSTEMS OF LINEAR EQUATIONS | Alexander Knop 18