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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
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