|Math 231|Section 1.1|Page 1
Chapter 1: Systems of Linear Equations and Matrices
• Section 1.1: Introduction to Systems of Linear Equations.
• Section 1.2: Gaussian Elimination.
• Section 1.3: Matrices and Matrix Operations.
• Section 1.4: Inverses; Algebraic Properties of Matrices.
• Section 1.5: Elementary Matrices and a Method for Finding.
• Section 1.6: More on Linear Systems and Invertible Matrices.
• Section 1.7: Diagonal, Triangular, and Symmetric Matrices.
, |Math 231|Section 1.1|Page 2
Section 1.1: Introduction to Systems of Linear Equations1
Concepts:
• Linear Equation
• Homogeneous linear equation
• System of linear equations
• Solution of a linear systems
• Consistent linear system
• Inconsistent linear system
• Parameter
• Parametric equations
• Augmented matrix
• Elementary row operations
Learning Outcomes.
After completing this section, you should be able to:
• Determine whether a given equation is linear.
• Determine whether a given set of numbers is a solution of a linear system.
• Find the augmented matrix of a linear system.
• Find the linear system corresponding to a given augmented matrix.
• Perform elementary row operations on a linear system.
• Determine whether a linear system is consistent or inconsistent.
• Find the set of solutions to a consistent linear system.
1
The materials of these lecture notes are based on the textbook of the course.
, |Math 231|Section 1.1|Page 3
Linear Equations
A linear equation in the 𝑛𝑛 variables 𝑥𝑥1 , 𝑥𝑥2 , … , 𝑥𝑥𝑛𝑛 to be one that can be expressed in
the form
𝑎𝑎1 𝑥𝑥1 + 𝑎𝑎2 𝑥𝑥2 + ⋯ + 𝑎𝑎𝑛𝑛 𝑥𝑥𝑛𝑛 = 𝑏𝑏
where 𝑎𝑎1 , 𝑎𝑎2 , … , 𝑎𝑎𝑛𝑛 , 𝑏𝑏 are constants, and 𝑎𝑎1 , 𝑎𝑎2 , … , 𝑎𝑎𝑛𝑛 are not all zero.
Remark: In the linear equation, we note the following:
1- power of the variables is 1.
2- No products in the variables.
Homogeneous Linear Equation
In the special case where 𝑏𝑏 = 0, the above linear equation has the form
𝑎𝑎1 𝑥𝑥1 + 𝑎𝑎2 𝑥𝑥2 + ⋯ + 𝑎𝑎𝑛𝑛 𝑥𝑥𝑛𝑛 = 0
which is called a homogeneous linear equation in the variables 𝑥𝑥1 , 𝑥𝑥2 , … , 𝑥𝑥𝑛𝑛 .
Examples of linear equations:
Examples of non-linear equations:
System of linear equations
A finite set of linear equations is called a system of linear equations; or, more briefly, a
linear system. The variables are called unknowns.
, |Math 231|Section 1.1|Page 4
Examples of linear systems
Remark.
A general linear system of 𝑚𝑚 equations in the 𝑛𝑛 unknowns 𝑥𝑥1 , 𝑥𝑥2 , … , 𝑥𝑥𝑛𝑛 can be written
as:
𝑎𝑎11 𝑥𝑥1 + 𝑎𝑎12 𝑥𝑥2 + ⋯ + 𝑎𝑎1𝑛𝑛 𝑥𝑥𝑛𝑛 = 𝑏𝑏1
𝑎𝑎21 𝑥𝑥1 + 𝑎𝑎22 𝑥𝑥2 + ⋯ + 𝑎𝑎2𝑛𝑛 𝑥𝑥𝑛𝑛 = 𝑏𝑏2
⋮ ⋮ ⋮ = ⋮
𝑎𝑎𝑚𝑚1 𝑥𝑥1 + 𝑎𝑎𝑚𝑚2 𝑥𝑥2 + ⋯ + 𝑎𝑎𝑚𝑚𝑚𝑚 𝑥𝑥𝑛𝑛 = 𝑏𝑏𝑚𝑚
Solution of a linear systemin
A solution of a linear systemin 𝑛𝑛 unknowns 𝑥𝑥1 , 𝑥𝑥2 , … , 𝑥𝑥𝑛𝑛 is a sequence of 𝑛𝑛 numbers
𝑠𝑠1 , 𝑠𝑠2 , … , 𝑠𝑠𝑛𝑛 for which the substitution
𝑥𝑥1 = 𝑠𝑠1 , 𝑥𝑥2 = 𝑠𝑠2 , … , 𝑥𝑥𝑛𝑛 = 𝑠𝑠𝑛𝑛
makes each equation a true statement.
Example. Verify that 𝑥𝑥 = 0 and 𝑦𝑦 = 3 is not a solution to the linear system
5𝑥𝑥 + 𝑦𝑦 = 3
2𝑥𝑥 − 𝑦𝑦 = 4
Example. Verify that 𝑥𝑥1 = 1, 𝑥𝑥2 = 2, and 𝑥𝑥3 = −1 is a solution to the linear system
4𝑥𝑥1 − 𝑥𝑥2 + 3𝑥𝑥3 = −1
3𝑥𝑥1 + 𝑥𝑥2 + 9𝑥𝑥3 = −4
Chapter 1: Systems of Linear Equations and Matrices
• Section 1.1: Introduction to Systems of Linear Equations.
• Section 1.2: Gaussian Elimination.
• Section 1.3: Matrices and Matrix Operations.
• Section 1.4: Inverses; Algebraic Properties of Matrices.
• Section 1.5: Elementary Matrices and a Method for Finding.
• Section 1.6: More on Linear Systems and Invertible Matrices.
• Section 1.7: Diagonal, Triangular, and Symmetric Matrices.
, |Math 231|Section 1.1|Page 2
Section 1.1: Introduction to Systems of Linear Equations1
Concepts:
• Linear Equation
• Homogeneous linear equation
• System of linear equations
• Solution of a linear systems
• Consistent linear system
• Inconsistent linear system
• Parameter
• Parametric equations
• Augmented matrix
• Elementary row operations
Learning Outcomes.
After completing this section, you should be able to:
• Determine whether a given equation is linear.
• Determine whether a given set of numbers is a solution of a linear system.
• Find the augmented matrix of a linear system.
• Find the linear system corresponding to a given augmented matrix.
• Perform elementary row operations on a linear system.
• Determine whether a linear system is consistent or inconsistent.
• Find the set of solutions to a consistent linear system.
1
The materials of these lecture notes are based on the textbook of the course.
, |Math 231|Section 1.1|Page 3
Linear Equations
A linear equation in the 𝑛𝑛 variables 𝑥𝑥1 , 𝑥𝑥2 , … , 𝑥𝑥𝑛𝑛 to be one that can be expressed in
the form
𝑎𝑎1 𝑥𝑥1 + 𝑎𝑎2 𝑥𝑥2 + ⋯ + 𝑎𝑎𝑛𝑛 𝑥𝑥𝑛𝑛 = 𝑏𝑏
where 𝑎𝑎1 , 𝑎𝑎2 , … , 𝑎𝑎𝑛𝑛 , 𝑏𝑏 are constants, and 𝑎𝑎1 , 𝑎𝑎2 , … , 𝑎𝑎𝑛𝑛 are not all zero.
Remark: In the linear equation, we note the following:
1- power of the variables is 1.
2- No products in the variables.
Homogeneous Linear Equation
In the special case where 𝑏𝑏 = 0, the above linear equation has the form
𝑎𝑎1 𝑥𝑥1 + 𝑎𝑎2 𝑥𝑥2 + ⋯ + 𝑎𝑎𝑛𝑛 𝑥𝑥𝑛𝑛 = 0
which is called a homogeneous linear equation in the variables 𝑥𝑥1 , 𝑥𝑥2 , … , 𝑥𝑥𝑛𝑛 .
Examples of linear equations:
Examples of non-linear equations:
System of linear equations
A finite set of linear equations is called a system of linear equations; or, more briefly, a
linear system. The variables are called unknowns.
, |Math 231|Section 1.1|Page 4
Examples of linear systems
Remark.
A general linear system of 𝑚𝑚 equations in the 𝑛𝑛 unknowns 𝑥𝑥1 , 𝑥𝑥2 , … , 𝑥𝑥𝑛𝑛 can be written
as:
𝑎𝑎11 𝑥𝑥1 + 𝑎𝑎12 𝑥𝑥2 + ⋯ + 𝑎𝑎1𝑛𝑛 𝑥𝑥𝑛𝑛 = 𝑏𝑏1
𝑎𝑎21 𝑥𝑥1 + 𝑎𝑎22 𝑥𝑥2 + ⋯ + 𝑎𝑎2𝑛𝑛 𝑥𝑥𝑛𝑛 = 𝑏𝑏2
⋮ ⋮ ⋮ = ⋮
𝑎𝑎𝑚𝑚1 𝑥𝑥1 + 𝑎𝑎𝑚𝑚2 𝑥𝑥2 + ⋯ + 𝑎𝑎𝑚𝑚𝑚𝑚 𝑥𝑥𝑛𝑛 = 𝑏𝑏𝑚𝑚
Solution of a linear systemin
A solution of a linear systemin 𝑛𝑛 unknowns 𝑥𝑥1 , 𝑥𝑥2 , … , 𝑥𝑥𝑛𝑛 is a sequence of 𝑛𝑛 numbers
𝑠𝑠1 , 𝑠𝑠2 , … , 𝑠𝑠𝑛𝑛 for which the substitution
𝑥𝑥1 = 𝑠𝑠1 , 𝑥𝑥2 = 𝑠𝑠2 , … , 𝑥𝑥𝑛𝑛 = 𝑠𝑠𝑛𝑛
makes each equation a true statement.
Example. Verify that 𝑥𝑥 = 0 and 𝑦𝑦 = 3 is not a solution to the linear system
5𝑥𝑥 + 𝑦𝑦 = 3
2𝑥𝑥 − 𝑦𝑦 = 4
Example. Verify that 𝑥𝑥1 = 1, 𝑥𝑥2 = 2, and 𝑥𝑥3 = −1 is a solution to the linear system
4𝑥𝑥1 − 𝑥𝑥2 + 3𝑥𝑥3 = −1
3𝑥𝑥1 + 𝑥𝑥2 + 9𝑥𝑥3 = −4
