Linear Algebra
        Lecture 3 - Linear Independence and Bases
              Oskar Kędzierski
              24 October 2022
                                                                                                                                                                                                                                                                                                                                                     DeĄnition
                                                                                                                                                                                                                                                                                                                                                     Vectors v1,:::,vk P V are said to be linearly dependent if there
                                                                                                                                                                                                                                                                                                                                                     exist real numbers α ,:::,α , not all of which are 0 such that
                                                                                                                                                                                                                                                                                                                                                                     1     k
                                                                                                                                                                                                                                                                                                                                                                      α1v1 `:::`αkvk “ 0:
                                                                                                                                                                                                                                                                                                                                                     Vectors v1,:::,vk P V are said to be linearly independent if they
                                                                                                                                                                                                                                                                                                                                                     are not linearly dependent.
                                                                                                                                                                                                                                                                                                                                                     By deĄnition, vectors v ,:::,v are linearly independent if
                                                                                                                                                                                                                                                                                                                                                                       1     k
                                                                                                                                                                                                                                                                                                                                                     α1v1 `:::`α v “0 implies that α1 “ ::: “ α “ 0.
                                                                                                                                                                                                                                                                                                                                                                 k k                       k
                                                                                                                                                                                                                                                                                                                                                     Linear independence does not depend on the order of vectors hence
                                                                                                                                                                                                                                                                                                                                                     we may talk about independent (Ąnite) sets. We assume that
                                                                                                                                                                                                                                                                                                                                                     empty set is linearly independent.
  Linearly (In)dependent Vectors
     Let V be a vector space.
                                                                                                                                                                                                                                                                                                                                                     Vectors v1,:::,vk P V are said to be linearly independent if they
                                                                                                                                                                                                                                                                                                                                                     are not linearly dependent.
                                                                                                                                                                                                                                                                                                                                                     By deĄnition, vectors v ,:::,v are linearly independent if
                                                                                                                                                                                                                                                                                                                                                                       1     k
                                                                                                                                                                                                                                                                                                                                                     α1v1 `:::`α v “0 implies that α1 “ ::: “ α “ 0.
                                                                                                                                                                                                                                                                                                                                                                 k k                       k
                                                                                                                                                                                                                                                                                                                                                     Linear independence does not depend on the order of vectors hence
                                                                                                                                                                                                                                                                                                                                                     we may talk about independent (Ąnite) sets. We assume that
                                                                                                                                                                                                                                                                                                                                                     empty set is linearly independent.
  Linearly (In)dependent Vectors
     Let V be a vector space.
     DeĄnition
     Vectors v1,:::,vk P V are said to be linearly dependent if there
     exist real numbers α1,:::,αk, not all of which are 0 such that
                      α v `:::`α v “0:
                       1 1       k k
                                                                                                                                                                                                                                                                                                                                                     By deĄnition, vectors v1,:::,vk are linearly independent if
                                                                                                                                                                                                                                                                                                                                                     α1v1 `:::`α v “0 implies that α1 “ ::: “ α “ 0.
                                                                                                                                                                                                                                                                                                                                                                 k k                       k
                                                                                                                                                                                                                                                                                                                                                     Linear independence does not depend on the order of vectors hence
                                                                                                                                                                                                                                                                                                                                                     we may talk about independent (Ąnite) sets. We assume that
                                                                                                                                                                                                                                                                                                                                                     empty set is linearly independent.
  Linearly (In)dependent Vectors
     Let V be a vector space.
     DeĄnition
     Vectors v1,:::,vk P V are said to be linearly dependent if there
     exist real numbers α1,:::,αk, not all of which are 0 such that
                      α v `:::`α v “0:
                       1 1       k k
     Vectors v1,:::,vk P V are said to be linearly independent if they
     are not linearly dependent.