Linear Algebra

Section I. Isomorphisms 157The map f 2 is onto because any member a + bx + cx 2 of the codomain is theimage of some member of the domain, namely it is the image of cx + ay + bz.For instance, 2 + 3x − 4x 2 is f 2 (−4x + 2y + 3z).The computations for structure preservation are like those in the prior example.This map preserves additionf 2((c1 x + c 2 y + c 3 z) + (d 1 x + d 2 y + d 3 z) )and scalar multiplication.= f 2((c1 + d 1 )x + (c 2 + d 2 )y + (c 3 + d 3 )z )= (c 2 + d 2 ) + (c 3 + d 3 )x + (c 1 + d 1 )x 2= (c 2 + c 3 x + c 1 x 2 ) + (d 2 + d 3 x + d 1 x 2 )= f 2 (c 1 x + c 2 y + c 3 z) + f 2 (d 1 x + d 2 y + d 3 z)f 2(r · (c1 x + c 2 y + c 3 z) ) = f 2 (rc 1 x + rc 2 y + rc 3 z)Thus f 2 is an isomorphism and we write V ∼ = P 2 .= rc 2 + rc 3 x + rc 1 x 2= r · (c 2 + c 3 x + c 1 x 2 )= r · f 2 (c 1 x + c 2 y + c 3 z)We are sometimes interested in an isomorphism of a space with itself, calledan automorphism. An identity map is an automorphism. The next two examplesshow that there are others.1.6 Example A dilation map d s : R 2 → R 2 that multiplies all vectors by anonzero scalar s is an automorphism of R 2 .d 1.5(⃗u)⃗u⃗vd 1.5−→d 1.5 (⃗v)A rotation or turning map t θ : R 2 → R 2 that rotates all vectors through an angleθ is an automorphism.⃗ut π/6−→t π/6 (⃗u)A third type of automorphism of R 2 is a map f l : R 2 → R 2 that flips or reflectsall vectors over a line l through the origin.