Homogeneous Equations Revisited      Definitions       Criteria for (in)dependence   Independence versus Dependence
                        Linear Dependence and Independence
                                                     A. Havens
                                            Department of Mathematics
                                      University of Massachusetts, Amherst
                                                February 7, 2018
                                                A. Havens      Linear Dependence and Independence
   Homogeneous Equations Revisited      Definitions       Criteria for (in)dependence   Independence versus Dependence
   Outline
          1 Homogeneous Equations Revisited
                  ANewPerspective on Ax = 0
                  Dependency Relations from Nontrivial Solutions
          2 Definitions
                  Non-triviality and Dependence
                  Linear Independence
          3 Criteria for (in)dependence
                  Special Cases in Low Dimensions
                  The Theory of Independence in ≥ 3 Variables
          4 Independence versus Dependence
                  Essential Ideas of linear (in)dependence
                                                A. Havens      Linear Dependence and Independence
   Homogeneous Equations Revisited      Definitions       Criteria for (in)dependence   Independence versus Dependence
   A New Perspective on Ax = 0
   Nontrivial Linear Combinations Equal to 0
          Reconsider the meaning of a nontrivial solution x 6= 0 ∈ Rn to a
          homogeneous system
                                                 Ax=0∈Rm.
          If the columns of A are a ,...,a , then this means there is a linear
                                                1          n
          combination
                                           x a +...+x a =0,
                                             1 1                n n
          where at least one of the coefficients x is nonzero.
                                                                    i
          Now, this could be uninteresting (e.g. if the matrix A is full of only
          zeroes), but we know of examples of nontrivial solutions to
          nontrivial systems.
                                                A. Havens      Linear Dependence and Independence
   Homogeneous Equations Revisited      Definitions       Criteria for (in)dependence   Independence versus Dependence
   A New Perspective on Ax = 0
   An Example with Nontrivial Solutions
          For example, the system
                                         2 3 −1  x1 
                                         1 2           1  x =0
                                         4 5 −5  x2 
                                                                     3
                                        |         {z         }| {z }
                                                  A                 x
          has infinitely many solutions, corresponding to the line of
          intersection of the three planes with equations 2x1 + 3x2 − x3 = 0,
          x +2x +x =0, and 4x +5x −5x =0.
            1        2       3                   1         2        3
                                                A. Havens      Linear Dependence and Independence