3155-4-20

Let f: R²àR² be defined by f(x, y) = (ax-by, bx+ay) where a²+b² ≠ 0 .

a. Prove that f is a bijection

i. First, to prove f is an injection, given f(x,y) = g(v,w)

ii. We must show that (x,y) = (v,w)

iii. Let t, u Î R² with t=(x,y) and u=(v,w)

iv. Let f(t) = At where A=[ (a –b), (b a) ]

v. So we want to show that if At = Au then t = u

vi. The det A = a²+b² ≠ 0 and A^-1 = 1/(a²+b²) [ (a b), (-b a) ]

vii. So if At = Au then A^-1At = A^-1Au à t = u

viii. Therefore f is an injection

ix. Since we have A^-1 = f^-1 then " y Î R² $ x Î R² = f^-1(y) such that f(x) = y

x. Therefore f is a surjection, and therefore a bijection as well

b. Find a formula for f^-1

i. f^-1(x) = A^-1t above

c. Give a geometric interpretation of f for the case a²+b²=1. (describe the effect f has on a geometric figure in the plane)

i. f(x,y) or At rotates the vector t around the origin counterclockwise by q where a = cos q and b = sin q