1.9 The Matrix of a Linear Transformation
                                 Math 2331 – Linear Algebra
                           1.9 The Matrix of a Linear Transformation
                                                          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 / 9
                     1.9 The Matrix of a Linear Transformation       Definition Theorem
                    1.9 The Matrix of a Linear Transformation
                  Matrix Transformation: Identity Matrix
                  Linear Transformation: Generalized Result
                  Matrix of a Linear Transformation
                          Theorem
                          Examples
                                                                                        2
                                  Geometric Linear Transformations of R
        Jiwen He, University of Houston                    Math 2331, Linear Algebra                                        2 / 9
                     1.9 The Matrix of a Linear Transformation       Definition Theorem
     Identity Matrix
           Identity Matrix
           I   is an n × n matrix with 1’s on the main left to right diagonal
            n
           and 0’s elsewhere. The ith column of I is labeled e .
                                                                           n                     i
           Example
                                                                         1 0 0 
                                   I   = e e e =0 1 0 
                                     3           1      2      3
                                                                              0 0 1
           Note that                                                              
                                                         1 0 0                  x
                                                                                  1
                                           I x =  0 1 0  x 
                                            3                                     2
                                                         0 0 1                  x
                                                                              3                
                =                        +                          +                          = .
        Jiwen He, University of Houston                    Math 2331, Linear Algebra                                        3 / 9
                     1.9 The Matrix of a Linear Transformation       Definition Theorem
     Linear Transformation
           Identity Matrix
           In general, for x in Rn,                  Inx =
           Linear Transformation
           From Section 1.8, if T : Rn → Rm is a linear transformation, then
           T(cu+dv)=cT(u)+dT(v).
           Generalized Result
                        T(c v +···+c v )=c T (v )+···+c T (v ).
                               1 1                  p p          1        1                  p        p
        Jiwen He, University of Houston                    Math 2331, Linear Algebra                                        4 / 9