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1.3 What are Linear Functions?                                                                  15


                   1.3     What are Linear Functions?


                   In calculus classes, the main subject of investigation was the rates of change
                   of functions. In linear algebra, functions will again be the focus of your
                   attention, but functions of a very special type. In precalculus you were
                   perhaps encouraged to think of a function as a machine f into which one
                   may feed a real number. For each input x this machine outputs a single real
                   number f(x).






















                      In linear algebra, the functions we study will have vectors (of some type)
                   as both inputs and outputs. We just saw that vectors are objects that can be
                   added or scalar multiplied—a very general notion—so the functions we are
                   going to study will look novel at first. So things don’t get too abstract, here
                   are five questions that can be rephrased in terms of functions of vectors.

                   Example 3 (Questions involving Functions of Vectors in Disguise)



                   (A) What number x satisfies 10x = 3?
                                                          
                                                 1           0
                                                 1
                                                             1 ?
                   (B) What 3-vector u satisfies  4    × u =   
                                                 0           1
                                                 R  1               R  1
                   (C) What polynomial p satisfies    p(y)dy = 0 and    yp(y)dy = 1?
                                                  −1                 −1
                   (D) What power series f(x) satisfies x  d  f(x) − 2f(x) = 0?
                                                       dx
                     4
                       The cross product appears in this equation.

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