Linear Algebra

268 Chapter Three. Maps Between SpacesTopic: Geometry of Linear MapsThe pictures below contrast f 1 (x) = e x and f 2 (x) = x 2 , which are nonlinear,with h 1 (x) = 2x and h 2 (x) = −x, which are linear. Each of the four picturesshows the domain R 1 on the left mapped to the codomain R 1 on the right.Arrows trace out where each map sends x = 0, x = 1, x = 2, x = −1, andx = −2. Note how the nonlinear maps distort the domain in transforming itinto the range. For instance, f 1 (1) is further from f 1 (2) than it is from f 1 (0) —the map is spreading the domain out unevenly so that an interval near x = 2 isspread apart more than is an interval near x = 0 when they are carried over tothe range.55550000-5-5-5-5The linear maps are nicer, more regular, in that for each map all of the domainis spread by the same factor.55550000-5-5-5-5The only linear maps from R 1 to R 1 are multiplications by a scalar. Inhigher dimensions more can happen. For instance, this linear transformation ofR 2 , rotates vectors counterclockwise, and is not just a scalar multiplication.