1, MAT 2355 September 23, 2015 This assignment is due Monday, September 28. Exercise 1. For each of the following functions f , determine if f is an isometry or not. If it is, prove it. If not, show a counterxample (e.g. two points P, Q such that d(P, Q) = d(f (P ), f (Q)). f : R3 R3 given by 0 sin() 0 1 0 0 sin() f (x, y, z) = cos() x y cos() z Where R is a xed constant. f : R2 R2 given by f (x, y) = 1/ 2 1/ 2 2 2 x y f : R2 R2 given by f (x, y) = (x + xy + 3, y 2 + x2 ). f : R2 R2 given by f (x, y) = cos(x ) sin(x ) x sin(x ) y cos(x ) Note that R>0 is a xed positive constant, while the x in the matrix is the rst coordinate of the vector (x, y). f : R2 R2 given by f (x, y) = (x + 3, y + 3) 2 (x + 3, y + 3) (1/ 2, 1/ 2) (1/ 2, 1/ 2) Note that within the square bracket we have the product of two vectors, which is a number. Of the functions above that are isometries, which preserve the scalar products u v? Exercise 2. In the next two problems we will see how to move between several equivalent denitions for a subspace of Rn . Note that each of the questions has multiple (innitely many) correct answers. 1 Consider the line l in R2 given by l = {u R2 | u (3, 4) = 25}. Find an equation in the form ax + by + c = 0 for l and nd vectors v, w such that l = {v + w | R}. Consider the plane H in R3 dened by H = {(x, y, z) | x + 2y + 4z 2 = 0}. Find three vectors u, v, w such that H = {u + v + w | , R}. Find a vector u and a number c such that H = {v R3 | u v = c}. Consider the straight line r in R3 dened by H = {u + v | R}, where u = (1, 1, 0), v = (1, 0, 1). Find two equations such that H = {(x, y, z) | ax + by + cz + d = 0 and a x+b y+ c z + d = 0}, and two vectors w, w and two numbers e, e such that H = {t R3 | w v = e and Exercise 3. w t=e} Let u, v be xed distinct vectors in R3 . Show that the set H = {w R3 | u w = v w } is a plane. Let u, v, w be xed distinct vectors in R3 , not belonging to the same line. Show that the set l = {w R3 | u w = vw = w w } is a straight line. Show that the vector u + (v u)/2 belongs to the plane H = {w R3 | u w = v w }. Show that if w is perpendicular to u v then w + u + (v u)/2 belongs to H = {w R3 | u w = v w }. Use this to write H as u + (v u)/2 + {w R3 | (u v) w = 0}. Exercise 4 (bonus exercise). Let u, v, w R3 be the vectors (1, 0, 0), (0, 1/2, 3/2), (0, 3/2, 1/2). Let O be the origin (0, 0, 0) in R3 . Show that if f, g are isometries such that f (u) = g(u), f (v) = g(v), f (w) = g(w) and f (O) = g(O) then f = g. Give an example where f (u) = g(u), f (v) = g(v), f (w) = g(w) but f is not equal to g. (Suggestion: consider the reection through the plane where f (u), f (v) and f (w) lie). Show that if f (u) = u, f (v) = v and f (O) = O then necessarily f is the identity map or the reection through the plane generated by u and v. 2