System of Linear Equations

A system of linear equation in the set of variables 𝒳= { x1, x2, … ,xn } is a set of linear equations

a 1⁢1 ⁢x1 + a 1⁢2 ⁢x2 +⋯+ a 1⁢n ⁢xn = b1 a 2⁢1 ⁢x1 + a 2⁢2 ⁢x2 +⋯+ a 2⁢n ⁢xn = b2 ⋮ a m⁢1 ⁢x1 + a m⁢2 ⁢x2 +⋯+ a m⁢n ⁢xn = bm

We call the above is standard representation of the system of linear equations.

The solution of a system of linear equations is defined

An n-tuple ( s1, s2, … ,sn )∈𝕂n is a solution to a system of linear equations if it is solution to each equation in the system of linear equations.

The above definition implies that the set of solutions to a system of linear equations is the intersection of the sets of solutions for each equation in the system of linear equations.

The tuple ( -1,1 ) is solutions to the system of linear equations

{ 5 x1 + 4⁢ x2 = -1 9⁢x1 + 7⁢ x2 = -2 -x1 - x2 = 0

The tuples ( -12,0 ,5,0 ) and ( -7,1 ,1,2 ) are both solutions to the system of linear equations

{ x3 + 2⁢ x4 = 5 x1 + 3⁢ x2 + 3⁢ x3 + 2⁢ x4 = 3 2⁢x1 + 6⁢ x2 + 5⁢ x3 + 2⁢ x4 = 1

For arbitrary constants s1, s2 each tuple computed as

x1 = 3 - 3⁢ s1 - 3⁢ s2 x2 = s1 x3 = 2 x4 = -1 x5 = s2

is a solution to

{ x1 + 3⁢ x2 + 3⁢ x3 + 2⁢ x4 + x5 = 7 3⁢x1 + 9⁢ x2 - 6⁢ x3 + 4⁢ x4 + 3⁢ x5 = -7 2⁢ x1 + 6⁢ x2 - 4⁢ x3 + 2⁢ x4 + 2⁢ x5 = -4

The solution set of the system of linear equations

{ 0⁢ x1 + 2⁢ x2 + x3 = 3 0⁢ x1 + 2⁢ x2 + x3 = 0

is empty. Even though the first equation has solution set

𝒮1= { ( s1 , -3-s3 2 , s3 ) | s1 , s3 ∈ 𝕂 } ≠∅

and the second has solution set

𝒮2= { ( t1 , t3 2 , t3 ) | t1 , t3 ∈ 𝕂 } ≠∅

their intersection is empty 𝒮1∩ 𝒮2=∅ . There is special name when the set of solutions is empty.

An system of linear equations is consistent if its solution set is non-empty; otherwise it is inconsistent.

Here is another inconsistent system of linear equations

{ x1 + 3⁢ x2 + 3⁢ x3 + 2⁢ x4 + x5 = 7 3⁢x1 + 9⁢ x2 - 6⁢ x3 + 4⁢ x4 + 3⁢ x5 = -7 2⁢ x1 + 6⁢ x2 - 4⁢ x3 + 2⁢ x4 + 2⁢ x5 = -4 0⁢ x1 + 0⁢ x2 + 0⁢ x3 + 0⁢ x4 + 0⁢ x5 = 7

For this particular system of linear equations, one of the equations itself has empty solution.

Homogeneous System of Linear Equations

An system of linear equations is homogeneous if each of its equations is homogeneous.

Any system of linear equations can be transformed to a homogenous system of linear equations by changing the constant of each linear equation to zero. Thus every system of linear equations has a corresponding system of linear equations.

The following system of linear equations is not homogeneous

{ 5 x1 + 4⁢ x2 = -1 9⁢x1 + 7⁢ x2 = -2 -x1 - x2 = 0

by susbsituting in each equation the constant with zero we obtain its corresponding system of linear equations:

{ 5 x1 + 4⁢ x2 = 0 9⁢x1 + 7⁢ x2 = 0 -x1 - x2 = 0

The homogeneous system of linear equations corresponding to the system of linear equations

{ x1 + 3⁢ x2 + 3⁢ x3 + 2⁢ x4 + x5 = 7 3⁢x1 + 9⁢ x2 - 6⁢ x3 + 4⁢ x4 + 3⁢ x5 = -7 2⁢ x1 + 6⁢ x2 - 4⁢ x3 + 2⁢ x4 + 2⁢ x5 = -4 0⁢ = 7

is

{ x1 + 3⁢ x2 + 3⁢ x3 + 2⁢ x4 + x5 = 0 3⁢x1 + 9⁢ x2 - 6⁢ x3 + 4⁢ x4 + 3⁢ x5 = 0 2⁢ x1 + 6⁢ x2 - 4⁢ x3 + 2⁢ x4 + 2⁢ x5 = 0 0 = 0

For any homogeneous system of linear equations the tuple consisting of all zeroes is necessarily a solution to that system of linear equations. Thus a homogeneous system of linear equations is always consistent.

Equivalent System of Linear Equations

Two system of linear equations are equivalent if they have the same set of solutions.

Consider following three system of linear equations: the first one being

{ x1 - 2⁢ x2 + 3⁢ x3 + x4 = -3 x1 - 2⁢ x2 + 3⁢ x3 + x4 = -3 2⁢ x1 - x2 + 3⁢ x3 - x4 = 0

the second one being

{ x1 - 2⁢ x2 + 3⁢ x3 + x4 = -3 2⁢ x1 - x2 + 3⁢ x3 - x4 = 0 2⁢ x1 - x2 + 3⁢ x3 - x4 = 0

and the third one

{ 2⁢ x1 - x2 + 3⁢ x3 - x4 = 0 x1 - 2⁢ x2 + 3⁢ x3 + x4 = -3 2⁢ x1 - x2 + 3⁢ x3 - x4 = 0

They all have the same set of solutions

x1 = 1 - s3 + s4 x2 = 2 + s3 + s4 x3 = s3 x4 = s4

and thus they are equivalent. However the system of linear equation

{ x1 - 2⁢ x2 + 3⁢ x3 + x4 = -3 2⁢ x1 - x2 + 3⁢ x3 - x4 = 0 x3 = 0 x4 = 0

while it may ``look'' related is not equivalent to any of them as it has a unique solution ( 1 ,2 ,0 ,0 ) .

Consider following two system of linear equations:

{ x1 - 2⁢ x2 + 3⁢ x3 + x4 = -3 2⁢ x1 - x2 + 3⁢ x3 - x4 = 0

and

{ y1 - 2⁢ y2 + 3⁢ y3 + y4 = -3 2⁢ y1 - y2 + 3⁢ y3 - y4 = 0

They are trivially equivalent as substituting the xi's or yi's with the same constants do not affect the computations.

The last example illustrates that as far as set of solutions is concerned (which is what we will be interested in), information is carried by the coefficients and the constants of each equation; the variables' labels are irrelevant. We will therefore focus on representing system of linear equations via matrices and vectors.