INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS I

LINEAR EQUATION IN N VARIABLES

A linear equation in n variables x₁, x₂, x₃, … x_n has the form :
a₁x₁ + a₂x₂ + a₃x₃ + … + a_nx_n = b

The Coefficients a₁, a₂, a₃, … a_n are real numbers
The Constant Term b is a real number
The Leading Coefficient is the number a₁
The Leading Variable is x₁

EXAMPLES OF LINEAR EQUATION
A.) 3x + 2y = 7
B.) 5x + y = 4
C.) 10a + 3b + 4c = 12

A Solution of a linear equation in n variables is a sequence of n real numbers s₁, s₂, s₃, … ,s_n arranged so the equation is satisfied when the values

 x₁ = s₁ , x₂ = s₂, x₃ = s₃, . . . , x_n = s_nare substituted into the equation
 

 The Solution set is the set of all solution of a linear equation, to describe the entire solution set of a linear equation. A Parametric Representation is often used

EXAMPLE OF A PARAMETRIC REPRESENTATION
Solve the Linear Equation: x + 2y = 4

* Solve for one of the variables on terms of the other variable
x = 4 – 2y

In this form, the variable y is Free, which means it can take on any real value
* Then represent the free variable as a Parameter by introducing a third variable

Let x₂ = v, V is an element all real numbers
x₁ = 4 – 2v
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~ Hachiman Hikigaya (My Teen Romantic Comedy Snafu)

INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS II

CONTINUATION...

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

a₁₁x₁ + a₁₂x₂ + a₁₃x₃ + . . . + a_1nx_n = b₁
a₂₁x₁ + a₂₂x₂ + a₂₃x₃ + . . . + a_2nx_n = b₂
a₃₁x₁ + a₃₂x₂ + a₃₃x₃ + . . . + a_3nx_n = b₃

a_m1x₁ + a_m2x₂ + a_m3x₃ + . . . + a_mnx_n = b_m

A system of linear equations is said to be Consistent if it has at least one solution and Inconsistent if it has no solution

EXAMPLES OF SYSTEMS OF TWO EQUATIONS IN TWO VARIABLES
Solve each system of linear equations 
A.) x + y = 3 and  x – y = -1 

The system has exactly one solution ( 1 , 2 ).The solution can be obtained by adding the two equations.*It is Consistent Independent since it has exactly one solution
B.) x + y = 3 and 2x + 2y = 6

The system has an infinite number of solutions. A parametric representation is shown as x = 3 – v, y = v, v is an element of all real numbers.*It is Consistent Dependent since it has an infinite number of solutions

C.) x + y = 3 and x + y = 1

The system has no solution because it is impossible for the sum of two numbers to be 1 and 3 simultaneously.*It is Inconsistent since it has no solution

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''They say people can change, but is that really true? If they decide they want to fly, will they grow wings? I don't think so. You don't change yourself. You change how you do things. You have to make your own way. You have to create a way to fly, even while you stay the same.''

~Sora (No Game , No Life)

INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS III

ANOTHER CONTINUATION ... 

A system is said to be in Row-Echelon Form if it follows a stair-step pattern and has leading coefficients of 1.One can easily solve a system using Back-substitution.

Two systems of Linear Equations are called equivalent if they precisely have the same solution set.

Operations that lead to equivalent systems of equations
1. Interchange two equations
2. Multiply an equation by a nonzero constant
3. Add a multiple of an equation to another equation

The process above is also called Gaussian Elimination, after the German Mathematician Carl Friedrich Gauss (1777-1855)

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''Math is like a puzzle, or a game. There's one answer, and you're going on an adventure to find it. But there's many differient paths you can take, so it's really rewarding when you reach the goal. Sometimes, it takes a bit of insight. Other times, you can craft your own method. Even the most convulted methods can get you to the answer. No matter how ugly your solution may be, the fact that you constructed something yourself is something to be proud of.''

                             

~Fujimiya Kaori (Ishuukan Friends)

GAUSSIAN ELIMINATION AND GAUSS JORDAN ELIMINATION I

THE MATRIX

A Matrix is a rectangular array consisting of m rows and n columns. If each entry of a matrix is a real number then the matrix is called a real matrix.

The entry a_ij, of the matrix is a number. The index i is called the Row subscript because it identifies the row in which the entry lies while the index j is called the Column subscript because it identifies the column which the entry lies.

A matrix with m rows and n columns is said to have a size of m x n. If m = n, then the Matrix is called a Square Matrix or Square of order n. For a square matrix, the entries a₁₁, a₂₂, a₃₃, … are called the Main Diagonal Entries.

A common application of matrices is to represent systems of linear equations. A matrix derived from the coefficients and constant terms is called Augmented Matrix. A matrix containing only the coefficients is called Coefficient Matrix.
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''There is no such thing as useless effort .''

                                   

~Kagami Taiga (Kuroko no Basket)

GAUSSIAN ELIMINATION AND GAUSS-JORDAN ELIMINATION II

CONTINUATION ...

Elementary Row Operations
1. Interchange two rows
2. Multiply a row by a nonzero constant
3. Add a multiple of a row to another row

Two matrices are said to be Row Equivalent of one can be obtained from the other by a finite sequence of elementary row operations

Definition of Row Echelon Form of a Matrix
1. All rows consisting entirely of zeros occur at the bottom of the matrix
2. For each row that does not consist entirely of zeros the first nonzero entry is
3. For two successive (nonzero) rows the leading 1 in the higher row is farther to the left that the leading 1 in the lower row 

A Homogeneous System is a system of linear equations in which each of the constant terms is zero

A homogeneous system must have at least one solution. A solution in which all of the variables have the value zero is called trivial or obvious.

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~ Yukino Yukinoshita (My Teen Romantic Comedy Snafu)

GAUSSIAN ELIMINATION AND GAUSS-JORDAN ELIMINATION III

ANOTHER CONTINUATION ...

Steps of Gaussian Elimination with Back-Substitution
1 Write the Augmented Matrix of the system of Linear Equations
2. Use Elementary row operations to rewrite the augmented matrix in row-echelon form
3. Write the system of Linear equations corresponding to the matrix in row-echelon form and use back substitution to find the solution

Gauss-Jordan Elimination
The second method of Elimination is named after Carl Gauss and Wilhelm Jordan. It continues the reduction process until a reduced row-echelon form is obtained.

A matrix in row-echelon form is in Reduced-row echelon form if every column that has a leading 1 has zeros in every position above and below its leading 1
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''Effort is nothing but a means to an end. Being proud of the fact you tried is just putting the carriage before the horse. ''

~ Akira Yoshii (Baka and Test)

NANANA'S BURIED TREASURE 1

Treasure: Skill of elimination:The Gaussian Way 

Is that my treasure?

Solve the following system of equations using Gaussian elimination.

*In augmented matrix form we have

* We now use the method of Gaussian Elimination

*We could proceed to try and replace the first element of row 2 with a zero, but we can actaully stop. To see why, convert back to a system of equations:---      NO SOLUTION

 
Special Thanks to the author of:

http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/gauss/gauss.html
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‘’Humankind cannot gain anything without first giving something in return. To obtain, something of equal value must be lost.’
’
~Alphonse Elric (Fullmetal Alchemist) 

NANANA'S BURIED TREASURE 2

Treasure: Elimination Skill: Gauss Jordan Combo

Is that supposed to be a joke?

EXAMPLE OF GAUSS-JORDAN ELIMINATION



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                                                               R.I.P. SCHEELE
                                                        episode 6, Akame ga Kill

  

APPLICATIONS OF SYSTEMS OF LINEAR EQUATIONS

The procedure Polynomial Curve Fitting is used when a collection of data is represented by n points in xy plane.

                              (x₁,y₁) , (x₂,y₂), … (x_n,y_n)

And you are asked to find a polynomial function of degree n-1 whose graph passes through the specified points

                                       P(x) = a₀ +a₁x +a₂x² + … + a_n-1x^n-1

Another application is the network analysis,

A Network is composed of branches and junctions which are used to analyze the flow by using system of linear equations. The total flow in a junction is equal to the total flow out of a junction.

Kirchhoff’s Law
1. All the current flow into a junction must flow out of it
2. The sum of all voltages is equal to the sum of the products of the current and resistance
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‘’You must live long as you are still strong. Never look at your life as something insignificant’’

                                  

~ Natsu Dragneel  (Fairy Tail)