Matriks Dan Ruang Vektor : Sistem Persamaan Liniear Dan Representasi Matriks

Matriks dan Ruang Vektor : Sistem Persamaan Liniear dan Representasi Matriks

Introduction to Systems of Linear Equations

Linear Equation

Definition of Linear Equation is a linear equation in the 𝑛 variables 𝑥1,𝑥2,…,𝑥𝑛 to be one that can be expressed in the form

𝑎1 𝑥1 + 𝑎2 𝑥2 + ⋯+ 𝑎𝑛 𝑥𝑛 = 𝑏

Where 𝑎1,𝑎2,…,𝑎𝑛 and 𝑏 are constants (the 𝑎′s are not all zero). There is a special case, which 𝑏 = 0 called homogeneous linear equation. A linear equation with two variables called a line. However, for three variables called a plane.

For Example :

As you can see the linear equation has exactly single type variable ( which not squared, root, nor higher ) for each variable that used in linear equation. Different from linear equation, non-linear equation, however, it has more than one type of variable with consists squared, root, and trigonometry sequences. Those sequences wasn't classified as linear equation and cannot be expressed as matrix form.

System of Linear Equation

Finite set of linear equation is system of linear equations (a linear system )

For example :

A general linear system of 𝑚 equations in the 𝑛 unknowns 𝑥1,𝑥2,…,𝑥𝑛 can be written as

Definition of Solution in Linear Algebra Sequence

Solution is a sequence of 𝑛 numbers 𝑠1,𝑠2,…,𝑠𝑛 for which the substitution 𝑥1 = 𝑠1,𝑥2 = 𝑠2,…,𝑥𝑛 = 𝑠𝑛 makes each equation a true statement

Exercise 1

Proof those equation that already has the solution with right answer

The three possible types of solution

1. No solution ( linear system is inconsistent ),

ex
𝑥 + 𝑦 = 4
3𝑥 + 3𝑦 = 6

2. Exactly one solution ( linear system is consistent )

ex
𝑥 − 𝑦 = 1
2𝑥 + 𝑦 = 6

3. Infinitely many solution ( linear system is consistent )

ex
4𝑥 − 2𝑦 = 1
16𝑥 − 8𝑦 = 4

Augmented Matrices

We can abbreviate the system by writing only the rectangular array of numbers

The matrix is called the augmented matrix

Elementary Row Operations

Basic method for solving linear system: perform algebraic operations that do not alter the solution set

• The algebraic operations: 
• Multiply an equation through by a nonzero constant 
• Interchange two equations 
• Add a constant times one equation to another 

• These three operations correspond to elementary row operations on a matrix: 
• Multiply a row through by a nonzero constant 
• Interchange two rows
• Add a constant times one row to another

Using Elementary Row Operations

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