Linear Algebra

Section I. Isomorphisms 161The map f 2 is onto because any member a + bx + cx 2 of the codomain isthe image of a member of the domain, namely 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 priorexample. 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)Every space is isomorphic to itself under the identity map.1.6 Definition An automorphism is an isomorphism of a space with itself.1.7 Example A dilation map d s : R 2 → R 2 that multiplies all vectors by a nonzeroscalar 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 anangle θ 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.f l (⃗u)⃗uf l−→