Let A € R"Xm, n > m. a) Use the SVD of A to deduce the SVD of AT A. b) If m = n and A is full-rank, use a) to show that ||AT A||2 = ||A||| and K2(AT A) = K2(A)².

1.49. (!) Let f and g be functions from R to R. For the sum and product of f and g (see Definition 1.25), determine which statements below are true. If true, provide a proof; if false, provide a counterexample. a) If f and g are bounded, then f + g is bounded. b) If f and g are bounded, then fg is bounded. c) If f + g is bounded, then f and g are bounded. d) If fg is bounded, then f and g are bounded. e) If both f + g and fg are bounded, then f and g are bounded. Y

Let S = Þ(D), where D = {(u, v) : u² + v² ≤ 1, u ≥ 0, v ≥ 0} and Þ(u, v) = (2u + 1, u − v, 3u + v). (a) Calculate the surface area of S. (Express numbers in exact form. Use symbolic notation and fractions where needed.) area(S) = Is √6 2 (b) Evaluate (3x – 3y) ds. Hint: Use polar coordinates. (Express numbers in exact form. Use symbolic notation and fractions where needed.) π Incorrect (3x - 3y) ds = √6 (8+3π) 12