Math 2331 { Linear Algebra

Echelon FormsMath 2331 Linear Row Reduction and Echelon FormsJiwen HeDepartment of Mathematics, University of jiwenhe/math2331 Jiwen He, University of HoustonMath 2331, Linear Algebra1 / Echelon FormsDefinition Reduction Solution Row Reduction and Echelon FormsEchelon Form and Reduced Echelon FormUniqueness of the Reduced Echelon FormPivot and Pivot ColumnRow Reduction AlgorithmReduce to Echelon Form (Forward Phase)then to REF (Backward Phase)Solutions of Linear SystemsBasic Variables and Free VariableParametric Descriptions of Solution SetsFinal Steps in Solving a Consistent Linear SystemBack-SubstitutionGeneral SolutionsExistence and Uniqueness TheoremUsing Row Reduction to Solve Linear SystemsConsistency QuestionsJiwen He, University of HoustonMath 2331, Linear Algebra2 / Echelon FormsDefinition Reduction Solution TheoremEchelon FormsEchelon Form (orRow Echelon Form)1 All nonzero rows are above any rows of all entry( left most nonzero entry) of a row is ina column to the right of the leading entry of the row above entries in a column below a leading entry are (Echelon forms)(a) 0 00 0 0 000 0 0 0 (b) 0 00 000 (c) 0 00 0 00 00 00 000 0 0 00 000 0 00 Jiwen He, University of HoustonMath 2331, Linear Algebra3 / Echelon Form

Example (Solutions of Linear Systems) 2 4 1 6 0 3 0 0 0 0 1 8 0 5 0 0 0 0 1 7 3 5 x 1 +6x 2 +3x 4 = 0 x 3 8x 4 = 5 x 5 = 7 pivot columns: basic variables: free variables: Jiwen He, University of Houston Math 2331, Linear Algebra 11 / 19

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