Representing Linear Maps with Matrices            Existence/Uniqueness Redux             Matrix Algebra
                    Linear Transformations and Matrix Algebra
                                                       A. Havens
                                              Department of Mathematics
                                        University of Massachusetts, Amherst
                                              February 10-16, 2018
                                                  A. Havens       Linear Transformations and Matrix Algebra
   Representing Linear Maps with Matrices            Existence/Uniqueness Redux             Matrix Algebra
   Outline
          1 Representing Linear Maps with Matrices
                   The Standard Basis of Rn
                   Finding Matrices Representing Linear Maps
          2 Existence/Uniqueness Redux
                   Reframing via Linear Transformations
                   Surjectivity, or Onto Maps
                   Injectivity, or One-To-One Maps
                   Theorems on Existence and Uniqueness
          3 Matrix Algebra
                   Composition of Maps and Matrix Multiplication
                   Matrices as Vectors: Scaling and Addition
                   Transposition
                                                  A. Havens       Linear Transformations and Matrix Algebra
   Representing Linear Maps with Matrices            Existence/Uniqueness Redux             Matrix Algebra
                           n
   The Standard Basis of R
   Components Revisited
          Observe that any x ∈ R2 can be written as a linear combination of
          vectors along the standard rectangular coordinate axes using their
          components relative to this standard rectangular coordinate
          system:                         ñ        ô          ñ      ô         ñ       ô
                                   x =        x1      =x         1      +x         0      .
                                              x            1     0           2     1
                                               2
          These two vectors along the coordinate axes will form the standard
          basis for R2.
                                                  A. Havens       Linear Transformations and Matrix Algebra
   Representing Linear Maps with Matrices            Existence/Uniqueness Redux             Matrix Algebra
                           n
   The Standard Basis of R
   Elementary Vectors
          Definition
          The vectors along the standard rectangular coordinate axes of R2
          are denoted                          ñ       ô                     ñ      ô
                                      e :=         1      ,        e :=         0      .
                                        1          0                 2          1
          They are called elementary vectors (hence the notation ei,
          i = 1,2), and the ordered list (e ,e ) is called the standard basis
                                                             1    2
          of R2.
          Observe that Span{e ,e } = R2.
                                            1     2
                                                  A. Havens       Linear Transformations and Matrix Algebra