1.1 Linear System
                                 Math 2331 – Linear Algebra
                                    1.1 Systems of Linear Equations
                                                          Jiwen He
                                  Department of Mathematics, University of Houston
                                                  jiwenhe@math.uh.edu
                                         math.uh.edu/∼jiwenhe/math2331
        Jiwen He, University of Houston                     Math 2331, Linear Algebra                                       1 / 19
                                           1.1 Linear System      Definition Fact Equivalence Matrix Reduction Consistency
                              1.1 Systems of Linear Equations
                  Basic Fact on Solution of a Linear System
                         Example: Two Equations in Two Variables
                         Example: Three Equations in Three Variables
                         Consistency
                         Equivalent Systems
                         Strategy for Solving a Linear System
                  Matrix Notation
                  Solving a System in Matrix Form by Row Eliminations
                         Elementary Row Operations
                         Row Eliminations to a Triangular Form
                         Row Eliminations to a Diagonal Form
                  Two Fundamental Questions
                         Existence
                         Uniqueness
        Jiwen He, University of Houston                  Math 2331, Linear Algebra                                    2 / 19
                                           1.1 Linear System      Definition Fact Equivalence Matrix Reduction Consistency
     Linear Equation
          ALinear Equation
                                        a x +a x +···+a x =b
                                          1 1        2 2                  n n
          Examples (Linear)
                                                                                  √
                      4x −5x +2=x                         and         x =2( 6−x )+x
                         1         2              1                     2                    1         3
                                    ↓                                                 ↓
                            rearranged                                         rearranged
                                    ↓                                                 ↓            √
                        3x −5x =−2                                     2x +x −x =2 6
                            1        2                                    1       2       3
          Examples (Not Linear)
                                                                                       √
                          4x1 −6x2 = x1x2                     and            x2 = 2 x1 −7
        Jiwen He, University of Houston                  Math 2331, Linear Algebra                                    3 / 19
                                           1.1 Linear System      Definition Fact Equivalence Matrix Reduction Consistency
     Linear System
          Asolution of a System of Linear Equations
          Alist (s ,s ,...,s ) of numbers that makes each equation in the
                       1    2        n
          system true when the values s ,s ,...,s are substituted for
                                                         1    2         n
          x1,x2,...,xn, respectively.
          Examples (Two Equations in Two Variables)
          Each equation determines a line in 2-space.
                     x1      + x2 = 10                                  x1     − 2x2 = −3
                  −x1        + x2 =                 0                 2x1      − 4x2 =                    8
                   one unique solution                                         no solution
        Jiwen He, University of Houston                  Math 2331, Linear Algebra                                    4 / 19