Lecture 2

LINEAR SYSTEM OF EQUATIONS

Learning outcomes: by the end of this lecture

1. You should know,

a)What is a linear system of equations

b)What is a homogeneous system

c)How to represent a linear system in matrix form

d) What is a coefficient matrix

e)What is an augmented matrix

2. You should be able to solve a linear system of

equations using:

a)Row operations:

i. Gauss-elimination method (REF)

ii. Gauss-Jordan method (RREF)

b) Inverse matrix method

Definition of a Linear Equation in n Variables:

A linear equation in n variable x1 , x2 ,…, xn has the form

a1x1+a2 x2+⋯+an xn=b

Where the coefficients a1 , a2 ,… ,an , b are real numbers (usually known). The number of a1 is the leading

coefficient and x1 is the leading variable. The collection of several linear equations is referred to as the system of linear equations.

Definition of System of m Linear Equation in n Variables:A system of m linear equations in n variables is a set of m equations, each of which is linear in the same n variables:

a11 x1+a12 x2+⋯+a1n xn=b1a21 x1+a22 x2+⋯+a2 n xn=b2 ⋮am 1 x1+am2 x2+⋯+amn xn=bm

where a ij ,b i , i=1,2 ,…,m , j=1 ,…n , are constants.

2x y

4x y 1x y

3x y

Example:1Consider the following system of linear equations:

x1−3x2+x3=43 x1+2 x2−5 x3=−2 ⇒ m=4 ,n=32 x1−x2+3 x3=8x1 −7 x3=6

Example: 2 Which of the following are linear equations?

Number of Solutions of a System of Linear Equations

Consider the following systems of linear equations

(a) (b) (c)

For a system of linear equations, precisely one of the following is true:(a) The system has exactly one solution.(b) The system has no solution.

(c) The system has infinitely many solutions. Consistent and InconsistentA system of linear equations is called consistent if it has at least one solution and inconsistent if it has no solution.

EquivalentTwo systems of linear equations are said to be equivalent if they have the same set of solutions.

Back – SubstitutionWhich of the following systems is easier to solve?

System (b) is said to be in row-echelon form. To solve such a system, use a procedure called back – substitution.

Augmented Matrices and Coefficient MatricesConsider the linear system

Let

is called the coefficient matrix of the system. is called the augmented matrix of the system. is called the constant matrix of the system.

It is possible to write the system

in the following matrix form

[a11 a12 ⋯ a1n

a21 a22 ⋯ a2n

⋮ ⋮ ⋮ ⋮am1 am2 ⋯ amn

] [ x1

x2

⋮xn

] = [b1

b2

⋮bm

] A X B

Example: 3 x− y=12 x -5y = 6

[3 −12 −5 ] [ x

y ] = [16 ] A ⋅ X = B

Row-EquivalentTwo matrices are said to be row-equivalent if one can be obtained by the other by a series of elementary row operations.