2. Let X ~ gamma(a, ). Prove that E(X) = o/1. la TA cin
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
1. Let a, b,cE Z be positive integers. Prove or disprove each of the following (a) If b | c, then gcd(a, b) gcd(a, c). (b) If b c, then ged(a., b) < gcd(a, c)
Exercise 2 (pts 5). Let g() E Z[2]. Prove that g(x) is irreducible over Zx if and only if g() is irreducible as polynomial in Q[o].
Problem 21.13. Fory E Z+, let Aj (L. . . have B CU-1Aj. Is B necessarily finite? Prove it or give a counterexample. ,j). Suppose that for some n E Z+, we
Problem 21.13. Fory E Z+, let Aj (L. . . have B CU-1Aj. Is B necessarily finite? Prove it or give a counterexample. ,j). Suppose that for some n E Z+, we
Problem 5. (10pts) Let A E Mm.n(k) be a matrix of rank 2. Prove that A can be written as B.C, where Be Mm.2(k) and CE M2.n(k).
Let x,y,z e Z. Prove that if x+y= 2, then at least one of , y, and z must be even.
T'he goal of this problem is to establish the following remarkable result: Bezout's theorern. If a, be Z50, then 3x, y є Z such that gcd(a, b) = ax + by. Here ged(a, b) denotes the greatest common divisor of a and b (i.e. the largest positive integer that divides both a and b). Throughout this problem, we'll use the notation (a) Write down five numbers that live in 2Z +3Z. What's a simpler name for the set 2Z +3Z?...
(3) If z = a + ib E C and |2| := Va² + b², prove that |zw| = |z||w]. Proof. Proof here. goes (4) Let y : C× → R* be defined by 9(z) = |z|. Use Problem (3) to prove that y is a homomorphism. Proof. Proof goes here.
2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n
2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n