Question

EXERCISE 3.55. Let V denote the ry-plane in R3, oriented by N : - (0,0, 1) function f: V V defined as orientation-reversing? Let V denote the yz-plane, oriented by N = (-1,0,0). Is the f(x, y, 0) = (0, y, a) orientation-preserving or

Answer 1

To check that the map is orientation preserving, we first need to find the matrix associated with the transformation and find its determinant If the determinant is positive then the map is orientation preserving else not.

Let B e1, 2, 3} denote the standard basis for R2.Then the matrix of f in the standard basis By 1, 2}, B (2, e3} is given by the action of fon these basis vectors as follows: f (e1) f(e2) _ f(1,0,0) (0,0, 1) 0 e21. e3 f(0, 1,0) (0, 1,0) 120 e3 To 1 (here we write cordinates as column vectors) 1 0

0 1 det =-1 <0 1 0 Therefore the function is not orientation preserving,