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Basic Concepts of Systems of Linear Equations

The Definition of a System of Linear Equations

The Definition of a Linear Equation

A linear equation in $n$ unknowns $x_1, x_2, \dots, x_n$ is an equation of the form: $a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = b$

where $a_{ij}$ are real numbers, $b_i$ is a real number, and $a_1, a_2, \dots, a_n$ are not all zero.

The Definition of a System of Linear Equations

A system of $m$ linear equations in $n$ unknowns $x_1, x_2, \dots, x_n$ :

\begin{cases} a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = b_1 \\ a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = b_2 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n = b_m \end{cases}

where $a_{ij}$ are real numbers and $b_i$ are real numbers.

Classification of Systems of Linear Equations

Homogeneous Linear Systems

If all the constant terms $b_i = 0$ in a system of linear equations, that is:

\begin{cases} a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = 0 \\ a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = 0 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n = 0 \end{cases}

then the system is called a homogeneous linear system.

Non-Homogeneous Linear Systems

If at least one constant term $b_i \neq 0$ occurs in a system of linear equations, then the system is called a non-homogeneous linear system.

Matrix Representation

The Coefficient Matrix

The coefficient matrix of the system of linear equations is:

A = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{pmatrix}

The Augmented Matrix

The augmented matrix of the system of linear equations is:

(A|\vec{b}) = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} & | & b_1 \\ a_{21} & a_{22} & \dots & a_{2n} & | & b_2 \\ \vdots & \vdots & \ddots & \vdots & | & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} & | & b_m \end{pmatrix}

The Matrix Form

Matrix Form

A system of linear equations can be written in matrix form: $A\vec{x} = \vec{b}$

where:

$A$ is the coefficient matrix
$\vec{x} = (x_1, x_2, \dots, x_n)^T$ is the vector of unknowns
$\vec{b} = (b_1, b_2, \dots, b_m)^T$ is the vector of constant terms

The Notion of a Solution

The Definition of a Solution

A set of values of the variables that satisfies every equation of the system is called a solution of the system.

The Solution Set

The set of all solutions of a system of linear equations is called the solution set of that system.

Classification of Solutions

Unique solution: the system has exactly one solution
Infinitely many solutions: the system has infinitely many solutions
No solution: the system has no solution

Equivalence of Systems of Linear Equations

The Definition of Equivalent Systems

If two systems of linear equations $A\vec{x} = \vec{b}$ and $C\vec{x} = \vec{d}$ have the same solution set, the two systems are called equivalent.

Equivalent Transformations

Applying any of the following transformations to a system of linear equations yields a new system equivalent to the original one:

Interchanging the positions of two equations
Multiplying an equation by a nonzero constant
Adding a multiple of one equation to another equation

Exercises

Exercise 1

Write the system of linear equations $\begin{cases} x + 2y = 5 \\ 3x - y = 4 \end{cases}$ in matrix form.

Reference Answer

system of linear equationsmatrix representationcriterion for solutionscoefficient matrix

Approach: Identify the coefficient matrix, the vector of unknowns and the vector of constant terms.

Steps:

The coefficient matrix: $A = \begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix}$
The vector of unknowns: $\vec{x} = \begin{pmatrix} x \\ y \end{pmatrix}$
The vector of constant terms: $\vec{b} = \begin{pmatrix} 5 \\ 4 \end{pmatrix}$
The matrix form: $\begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 4 \end{pmatrix}$

Answer: $A\vec{x} = \vec{b}$ , where $A = \begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix}$ , $\vec{x} = \begin{pmatrix} x \\ y \end{pmatrix}$ , $\vec{b} = \begin{pmatrix} 5 \\ 4 \end{pmatrix}$

Exercise 2

Decide whether the system $\begin{cases} x + y = 2 \\ 2x + 2y = 4 \end{cases}$ is homogeneous or non-homogeneous.

Reference Answer

system of linear equationsmatrix representationcriterion for solutionsconstant terms

Approach: Check whether all the constant terms are zero.

Steps:

The first equation: $x + y = 2$ , the constant term is 2, which is not zero
The second equation: $2x + 2y = 4$ , the constant term is 4, which is not zero
Neither equation has a zero constant term

Answer: it is a non-homogeneous linear system

Exercise 3

Write the augmented matrix of the system $\begin{cases} x + y + z = 0 \\ 2x + 3y + z = 0 \end{cases}$ .

Reference Answer

system of linear equationsmatrix representationcriterion for solutionsconstant terms

Approach: Arrange the coefficients and the constant terms into a matrix in order.

Steps:

$\begin{pmatrix} 1 & 1 & 1 & | & 0 \\ 2 & 3 & 1 & | & 0 \end{pmatrix}$

Answer: $(A|\vec{b}) = \begin{pmatrix} 1 & 1 & 1 & | & 0 \\ 2 & 3 & 1 & | & 0 \end{pmatrix}$

Exercise 4

Decide whether the system $\begin{cases} x + y = 1 \\ x + y = 2 \end{cases}$ has a solution.

Reference Answer

system of linear equationsmatrix representationcriterion for solutions

Approach: Analyse the relation between the two equations.

Steps:

The first equation: $x + y = 1$
The second equation: $x + y = 2$
The two equations have the same left-hand side but different right-hand sides
There are no $x, y$ that satisfy both equations at once

Answer: there is no solution

Exercise 5

Prove that: the homogeneous linear system $A\vec{x} = \vec{0}$ always has a solution.

Reference Answer

system of linear equationsmatrix representationcriterion for solutionszero vector

Approach: Use the properties of the zero vector.

Steps:

Let $\vec{x} = \vec{0} = (0, 0, \dots, 0)^T$
Substitute into the equation: $A\vec{0} = \vec{0}$
A matrix times the zero vector equals the zero vector
Therefore $\vec{x} = \vec{0}$ is a solution of the homogeneous system

Answer: the proof is complete; the zero solution always exists

The Definition of Linear Dependence

Linear dependence: let $\vec{v}_1, \vec{v}_2, \dots, \vec{v}_m$ be a system of vectors in a vector space. If there exist numbers $k_1, k_2, \dots, k_m$ that are not all zero such that: $k_1\vec{v}_1 + k_2\vec{v}_2 + \dots + k_m\vec{v}_m = \vec{0}$

then this system of vectors is called linearly dependent; otherwise they are called linearly independent.

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation / Note	Meaning in This Article
$x_1, x_2, \dots, x_n$	Mathematical symbol	x sub 1, x sub 2	The $n$ unknowns of the system
$a_{ij}$	Mathematical symbol	a sub i j	The entry in row $i$ , column $j$ of the coefficient matrix
$A$	Mathematical symbol	matrix A	The coefficient matrix, formed by all the coefficients $a_{ij}$
$(A\mid \vec{b})$	Mathematical symbol	augmented matrix	The augmented matrix: the coefficient matrix with the vector of constant terms appended
$A\vec{x} = \vec{b}$	Mathematical symbol	A x equals b	The matrix form of the system of linear equations
$\vec{x} = (x_1, x_2, \dots, x_n)^T$	Mathematical symbol	x vector	The vector of unknowns
$A\vec{x} = \vec{0}$	Mathematical symbol	A x equals zero	The homogeneous system (all constant terms zero)

Systems of Linear Equations

Gaussian Elimination

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Basic Concepts of Systems of Linear Equations

The Definition of a System of Linear Equations

The Definition of a Linear Equation

The Definition of a System of Linear Equations

Classification of Systems of Linear Equations

Homogeneous Linear Systems

Non-Homogeneous Linear Systems

Matrix Representation

The Coefficient Matrix

The Augmented Matrix

The Matrix Form

The Notion of a Solution

The Definition of a Solution

The Solution Set

Classification of Solutions

Equivalence of Systems of Linear Equations

The Definition of Equivalent Systems

Equivalent Transformations

Exercises

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

The Definition of Linear Dependence

Summary

Symbols Used in This Article

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