Linear Algebra

TopicGeometry of Linear MapsThe pictures below contrast the nonlinear maps f 1 (x) = e x and f 2 (x) = x 2 withthe linear maps h 1 (x) = 2x and h 2 (x) = −x. Each shows the domain R 1 on theleft mapped to the codomain R 1 on the right. Arrows trace where each mapsends x = 0, x = 1, x = 2, x = −1, and x = −2.Note how the nonlinear maps distort the domain in transforming it into therange. For instance, f 1 (1) is further from f 1 (2) than it is from f 1 (0) — the mapspreads the domain out unevenly so that in moving from domain to range aninterval near x = 2 spreads apart more than is an interval near x = 0.55550000The linear maps are nicer, more regular, in that for each map all of the domainspreads by the same factor.55550000-5-5-5-5