1、Elementary Row Operations行初等變換 ( Replacement ) Replace one row by the sum of itself and a multiple of another row.( Interchange ) Interchange two rows.( Scaling ) Multiply all entries in a row by a nonzero constant.替換 交換 倍乘 121002884599121002884599121001440013Exp1:12100288031391210014403139 01482321

2、5871014823215871232101480002.5Exp2:.2321014858712321014800.521 5.1.2 Row Reduction and Echelon Forms行化簡 與 階梯形 DEFINITION-echelon form ( or row echelon form ) 1）All nonzero rows are above any rows of all zeros.2）Each leading entry of a row is in a column to the right of the leading entry of the row a

3、bove it.先導元素3）All entries in a column below a leading entry are zeros.232101480002.51210014400134、The leading entry in each nonzero row is 15、Each leading 1 is the only nonzero entry in its column.Echelon matrix matrix that is in echelon form. 10029010160013232101480002.5232101480001 reduced echelon

4、 form (or reduced row echelon form) 簡化階梯形2320014000012010001400001105001400001 A nonzero matrix row reduced ( transformed by elementary row oprations) echelon form. A nonzero matrix row reduced ( transformed by elementary row oprations) unique reduced echelon form. THEOREM 1Each matrix is row equiva

5、lent to one and only one reduced echelon form.an echelon form of matrix A；an reduced echelon form of matrix A. The leading entries are always in the same positions in any echelon form obtained from a given matrix. DEFINITION A pivot position (主元位置) in a matrix A is a location in A that corresponds t

6、o a leading 1 in the reduced echelon form of A. A pivot column (主元列) is a column of A that contains a pivot position.The Row Reduced Algorithm 行簡化算法03664537858939129615134325012213036645391296153785890366453912961502442603664513432501221300001410230240122070000141343030122070000140364912131230311459

7、7145970246605101515036491459701233000000005010305012030001000000145971213123031036491459701233012330364914597012330001000000Solution of Linear Systems105101140000132351400 xxxxx1, x2 Basic variables 基本變量x3 free variables 自由變量132331 54.xxxxxis freeGeneral solution 通解 EXAMPLE Determine if the linear s

8、ystem is consistentwhich the augmented matrix is1625240028130000171242344563547xxxxis freexxxis freexParametric descriptions of solution sets 解集的參數表示 16250100028010000017160300001405000017 Existence and Uniqueness Questions 存在性和唯一性問題 THEOREM 2Existence and Uniqueness Theorem A linear system is consi

9、stent if and only if the rightmost column of the augmented matrix is not a pivot column. If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variables, or (ii) infinity many solutions, when there is at least one free variable.Using Ro

10、w Reduction to solve a linear system 1. Augmented matrix2. Echelon form3. Reduced echelon form4. The system of equations5. Rewrite27403212321321321xxxxxxxxxEx1：Find the general solutions of the linear system201174132121A221310310121021000310121023000310501023000310501.233533231freeisxxxxxEx2：Find the general so