Recent content by samkolb

Homework Statement If a, b are positive integers, b > 2, prove that 2^a + 1 is not divisible by 2^b - 1. Homework Equations prime factorization of integers The Attempt at a Solution Suppose (2^a + 1)/(2^b - 1) = x, x an integer. Then x + 1 = (2^a + 2^b)/(2^b - 1). Write x...

What if det P = i. Then det P + (det P)* = 0 (I'm assuming that x* means the complex conjugate of x). Also, if each entry of P is pure imaginary,then P + P* = 0. I was unable to derive the formula det Q = 1/2(det P + (det P)*. Computing det Q explicitly in terms of the entries in P I got...

Let A and B be 2x2 real matricies, and suppose there exists an invertible complex 2x2 matrix P such that B = [P^(-1)]AP. Show that there exists a real invertible 2x2 matrix Q such that B = [Q^(-1)]AQ. A and B are similar when thought of as complex matricies, so they represent the same...

I don't know if this helps, since P* may not be real.

Homework Statement Let A and B be 2x2 real matricies, and suppose there exists an invertible complex 2x2 matrix P such that B = [P^(-1)]AP. Show that there exists a real invertible 2x2 matrix Q such that B = [Q^(-1)]AQ. Homework Equations A and B are similar when thought of as...

An affine geometry is a nonempty set of points A, together with a set of lines, where each line connects one or more of the points in A. A collineation of A is a bijection f: A --> A that carries lines to lines. That is, if P,Q are points in A lying on the same line, then f(P), f(Q) are...

I think I get it now. Setting e = (a -b)/2 in (i) gives (a + b)/2 < xn for all n >= N ==> (a - b)/2 < xn - b for all n >= N ==> no subsequence of xn can converge to b, since (a - b)/2 > 0 ==> b is not a cluster point. thanks!

So I set e = a - b in (i) and get b < xn for all n>=N. I don't see the problem with this. I think this implies all cluster points x satisfy b <= x, which means b <= a, but now I'm back where I started.

This is part of a theorem which is left unproved in "Elementary Classical Analysis" by Marsden and Hoffman. Let xn be a sequence in R which is bounded below. Let a be in R. Suppose: (i) For all e > 0 there is an N such that a - e < xn for all n >= N. (ii) For all e > 0 and all M...

If I am given a propisition P(m,n) and asked to show that it is true for all integers m and n, how do I go about that? My strategy is to fix one of the variables, say m, and then proceed to use induction on n. Once I've shown that P(m,n) holds for all n when m is fixed, I then conclude that...

Thanks for the hint. That works. But why does my proof for n=2 not work for noncommutative rings? Since the only terms in the expansion of (x+x)^2 are powers of x, I don't think I ever used commutativity.

Homework Statement Let R be a ring and suppose there exists a positive even integer n such that x^n = x for every x in R. Show that -x = x for every x in R. Homework Equations The Attempt at a Solution I solved the case where n = 2. Let x be in R. (x+x)^2= x+x = 2x...

Homework Statement Show that in a PID, every ideal is contained in a maximal ideal. Hint: Use the Ascending Chain Condition for Ideals Homework Equations Every ideal in a PID is a principal ideal domain. If p is an irreducible element of a PID, then <p> is a maximal ideal. The...

I think I get it now. If P is not in some Ai, then Ai must exist in {A1, A2, ...}, which is empty. Thanks.

I'm starting a book on topology and I've come to this come to this confusing statement: Let A1, A2, ... be subsets of some universal set U. If the class {A1, A2, ...} is empty, then the intersection of all the Ai is U. I know that the intersection of empty sets is empty, but I don't...