1.7 Linear Independence
                                 Math 2331 – Linear Algebra
                                            1.7 Linear Independence
                                                          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 / 17
                                      1.7 Linear Independence     Definition Matrix Columns Special Cases
                                      1.7 Linear Independence
                  Linear Independence and Homogeneous System
                  Linear Independence: Definition
                  Linear Independence of Matrix Columns
                  Special Cases
                         ASet of One Vector
                         ASet of Two Vectors
                         ASet Containing the 0 Vector
                         ASet Containing Too Many Vectors
                  Characterization of Linearly Dependent Sets
                         Theorem: Linear Dependence and Linear Combination
        Jiwen He, University of Houston                  Math 2331, Linear Algebra                                     2 / 17
                                      1.7 Linear Independence     Definition Matrix Columns Special Cases
     Linear Independence and Homogeneous System
          Example
          Ahomogeneous system such as
                                      1 2 −3  x1                            0 
                                      3 5             9  x2  =  0 
                                          5 9          3          x3               0
          can be viewed as a vector equation
                                1                2                −3               0 
                           x  3 +x  5 +x  9 = 0 .
                             1                  2                  3
                                   5                  9                     3               0
          The vector equation has the trivial solution (x = 0, x = 0,
                                                                                   1            2
          x3 = 0), but is this the only solution?
        Jiwen He, University of Houston                  Math 2331, Linear Algebra                                     3 / 17
                                      1.7 Linear Independence     Definition Matrix Columns Special Cases
     Linear Independence: Definition
          Linear Independence
                                                                     n
          Aset of vectors {v ,v ,...,v } in R is said to be linearly
                                        1    2           p
          independent if the vector equation
                                        x v +x v +···+x v =0
                                          1 1         2 2                 p p
          has only the trivial solution.
          Linear Dpendence
          The set {v ,v ,...,v } is said to be linearly dependent if there
                           1     2           p
          exists weights c ,...,c ,not all 0, such that
                                   1           p
                                        c v +c v +···+c v =0.
                                         1 1         2 2                 p p
                                                               ↑
                                         linear dependence relation
                                       (when weights are not all zero)
        Jiwen He, University of Houston                  Math 2331, Linear Algebra                                     4 / 17