2005/7 Linear system-1 The Linear Equation System and Eliminations

2005/7Linear system-2 Linear equation system over F Coefficients a 1, a 2, a 3, …, a n  F and constant term b  F. a 1 is called a leading coefficient ( 領先係數 ) and x 1 is called a leading variable. 注意： (1) 線性方程式之變數不可以是相乘或是開根號，且 變數不能被包含在三角、指數或對數函數裡面。 (2) 變數只能以第一冪次的方程式表示 。

2005/7Linear system-3 Ex: Linear or Non-linear

2005/7Linear system-4 Linear combination of the column vectors of matrix A  Ax = x 1 A 1 + x 2 A 2 +  + x n A n

2005/7Linear system-5 The Solution of a Linear Equation System For a linear equation system, only one of the following statements will be true: (1) There is exactly one solution. (consistent) (2) There are infinitely many solutions. (consistent) (3) There is no solution. (inconsistent)

2005/7Linear system-6 Example: (1) (2) (3)

2005/7Linear system-7 Ex: Use back substitution to solve the linear equation system Sol: Let substitute to (1) The only solution is

2005/7Linear system-8 Equivalent If two linear equation systems have the exactly same solution sets, then we say they are equivalent to each other. The following operations will produce equivalent linear equation systems. (1) Exchange two equations. (2) Multiple a nonzero constant to an equation. (3) Add two equations.

2005/7Linear system-9 Ex: Solve the linear equation system. Sol:

2005/7Linear system-10 The solution is

2005/7Linear system-11 Ex: Solve the given system of linear equations. Sol: contradiction This system has no solution.

2005/7Linear system-12 A system of Linear Equations (or a linear equation system) over a field (real numbers R or complex number C) (matrix of coefficients) Axb

2005/7Linear system-13 The augmented matrix ( 增廣矩陣 ) The coefficient matrix ( 係數矩陣 )

2005/7Linear system-14 Three elementary row operations (1) 交換兩列 (2) 乘上一個非零常數到某列 (3) 一列的倍數加到另一列 row equivalent 若一矩陣可由另一矩陣的一些基本列運算來獲得，則 此兩個矩陣稱為列等價 (row equivalent)

2005/7Linear system-15 Ex: Elementary row operations

2005/7Linear system-16 The row-echelon form ( 列梯形形式 ) (1) 全部為零的列在矩陣最底下 (2) 不全為零的列，其第一個非零元素為 1 ，稱為領先 1 (leading 1) (3) 對兩相鄰的非零列而言，較高列之領先 1 出現在較 低列之領先 1 的左邊 The reduced row-echelon form ( 列簡梯形形式 ) (1) ~ (3) 同上 (4) 在領先 1 的那一行除了領先 1 以外的位置全部為零

2005/7Linear system-17 Ex: 判斷下列矩陣為列梯形形式或列簡梯形形式

2005/7Linear system-18 The Gaussian elimination ( 高斯消去法 ) 將矩陣化簡為列梯形形式的程序 The Gauss-Jordan elimination ( 高斯 - 喬登消去法 ) 將矩陣化簡為列簡梯形形式的程序 注意： (1) 每個矩陣只有一個列簡梯形形式 (2) 每個矩陣可以有很多種列梯形形式 ( 不同的列運算 會產生不同的列梯形形式 )

2005/7Linear system-19 最左邊的非零行 產生 leading 1 讓在 leading 1 下的元素為 0 leading 1 產生 leading 1 最左邊的非零行 Ex: 高斯消去法與高斯喬登消去法之步驟說明 子矩陣

2005/7Linear system-20 讓在 leading 1 下的元素為 0 讓 leading 1 以外的其他位置為 0 leading 1 最左邊的非零行 產生 leading 1 leading 1 子矩陣

2005/7Linear system-21 Ex: Use Gauss-Jordan elimination to solve the system of linear equations. Sol:

2005/7Linear system-22 Ex: Solve the linear equation system. Sol: Let This system has infinitely many solutions.

2005/7Linear system-23 The homogeneous system 若一線性方程系統的常數項均為零時， 則此系統為齊次系統

2005/7Linear system-24 Trivial solution of a homogeneous system ( 顯然解 ) Nontrivial solution( 非顯然解 ) 顯然解之外的其他解 注意： (1) 所有的齊次系統均為一致性 (consistent) 系統 (2) 若系統的方程式比變數少，則有無限多組解 (3) 對於一個齊次系統來說，下列有一為真 (a) 系統只有一個顯然解 (b) 系統除了顯然解外還有無限多組解 ( 任意 n 變數齊次系統的解 )

2005/7Linear system-25 Ex: Find the solution of the given homogeneous system. Sol: Let