1.1 Systems of Linear Equations

1.1 Systems of Linear Equations

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• linear equation

= 일차 방정식

ex)

x+2y = 3

-x+3y =2


(linear equation X

x^2+2y = 3 

x^2 + 3y = 2)

• Solution set

  	ex)
  
  	x + 2y = 3
  	3x + 6y = 9
  
  	x = 1 y = 1
  	x = 2 y = 1/2
  	...

{ (1, 1), (2, 1/2), ...} ⇒ solution set

• Equivalent

  ex)
  
  x + 2y = 3
  3x + 6y = 9
  
  x + 2y = 3 
  9x + 18y = 27

→ they’ve the same solution set

1. no solution (inconsistent)
2. one solution (consistent)
3. infinitely many solutions (consistent)
   참고

• Matrix Notation
  ex)
  x -2y + z = 0
  2y - 8z = 8
  5x -5z = 10

• Coefficient matrix

  [1 -2 1
   0 2 -8
   5 0 -5]

  
  
  

• argumented matrix

  [1 -2 1 0
   0 2 -8 8
   5 0 -5 10]

  
  ** cf)

  [1 2 3 4
   3 2 1 0
   3 1 3 5]
  
  → argumented - 3 variables 3 equations
  
  → coefficient - 3 variables 4 equations

• Solving a Linear System
  ex)
  $\begin{bmatrix} 1 & -2 & 1 & 0\\ 0 & 2 & -8 & 8\\ 5 & 0 & -5 & 10 \end{bmatrix}$
  equation 1 (-5) + equation 3 = new equation 3
  $\begin{bmatrix} 1 & -2 & 1 & 0\\ 0 & 2 & -8 & 8\\ 0 & 10 & -10 & 10 \end{bmatrix}$
  equation / 2
  $\begin{bmatrix} 1 & -2 & 1 & 0\\ 0 & 1 & -4 & 4\\ 0 & 10 & -10 & 10 \end{bmatrix}$
  equation 2 (-10) + equation 3 = new equation 3
  $\begin{bmatrix} 1 & -2 & 1 & 0\\ 0 & 1 & -4 & 4\\ 0 & 0 & 30 & -30 \end{bmatrix}$
  equation 3 / 30
  $\begin{bmatrix} 1 & -2 & 1 & 0\\ 0 & 2 & -8 & 8\\ 0 & 0 & 1 & -1 \end{bmatrix}$
  ⇒ x3 = -1, x2 = 0, x1 = 1 ⇒ (1,0,-1)
  
  

• ELEMENTARY ROW OPERATIONS
  기본 행렬
  1. Replacement
  2. Interchange
  3. Scaling
  
  

• row equivalent
  same solution
  not affect any other system
  
  

• Existence and Uniqueness Question

  • solution
    
    • Is the system consistent; that is, does at least one solution exist? → YES!
    • If a solution exists, is it the only one; that is, is the solution unique? → NO → one or infinitly many
      
      
  • Exercises
    x3 = 1/2 → no solution = Inconsistent