Recent content by spamiam

How would you parametrize the line segment between the points ##x## and ##y##? If you know how to do that, I think you'll see the answer.

Don't you mean ##\mathbb{R}/\mathbb{Z}## tiny-tim? ##\mathbb{N}## is not a group, so I'm not sure I know what it means for it to act on ##\mathbb{R}##... By the way, the notation ##\mathbb{R}/\mathbb{Z}## is somewhat ambiguous: we might mean that we are identifying all points in the subspace...

100% false. For one thing, an automorphism ##\psi## must fix the identity, that is ##\psi(1) = 1##, so an arbitrary permutation is not in general an automorphism. For instance, ##\text{Aut}(\mathbb{Z}/n\mathbb{Z}) = (\mathbb{Z}/n\mathbb{Z})^\times##, and...

Maybe the definition of permutation is what's causing your misunderstanding. A permutation on a set X is simply a bijection ##f : X \to X##. Often we deal with permutations of finite sets of positive integers, but this need not be the case. For instance, the function ##f: \mathbb{R} \to...

I think the usual method is the Todd-Coxeter algorithm. It's covered quite extensively in Artin's Algebra.

It's not that strange at all. It's just like a clock with only two hours: 0 and 1. Take a look at this article: http://en.wikipedia.org/wiki/Modular_arithmetic No, there are axioms that + and * must satisfy in order for (F, +, *) to be considered a field. Briefly, (F, +) must be an...

Use \left and \right before your parentheses and brackets to have them automatically resize. You can click quote to check out the example below: $$ \lim_{x \to 0} \left( \frac{(1+x)^{1/x}}{e} \right)^{1/x} $$ Your solution is correct (I used L'Hopital's Rule), but I don't see a way to do...

Hi joao! Thanks for the interesting problem! It took me a while to wrap my head around it. You've made good progress by showing that ##f(2^k) = 2^{k+1} - 1##. Now can you use rule III to compute ##f(2^k - 1)##? How about ##f(2^k - 3)##? How about ##f(2^k - m)## where ##m<2^{k-1}##? Then...

The eigenvalues look fine. What are the B-coordinates of a polynomial ##p(x) = ax^2 + bx +c##? How does the matrix ##[T]_B## transform those coordinates?

A good start would be to apply ##T## to the basis vectors ##1, x, x^2##. What do you know about ##[T]_B##? In particular, what are its columns?

Legal? It's a substitution. If I said "Let ##\theta = \arcsin\left(x/2\right)##," would you still take issue with it? Certainly there is a lot of work you'd have to do to provide a rigorous proof that it works, but I'm guessing you've done substitutions before without taking a real analysis...

robert, the method I outlined in my original response (which I've quoted above) is still correct. You had the right idea, but as Bohrok pointed out, the substitution should be ##x = 2 \sin \theta## (as I said originally). You can then use the identity ##1 - \sin^2 \theta = \cos^2 \theta##...

I'm guessing it's just a typical trig sub. Set ##x = 2 \sin \theta##, use a trigonometric identity, integrate and then solve for ##\theta## in terms of x. If you haven't seen this technique before, check out this link.

Good start. You're looking for the invariant factor decomposition of the group--let's start out by computing its elementary divisor decomposition. Use the Chinese remainder theorem again on ##\mathbb{Z}/18 \mathbb{Z} \times \mathbb{Z}/ 16 \mathbb{Z} \times \mathbb{Z}/ 22 \mathbb{Z}## to write...

Even if you copied it right, the hint as stated is wrong. It should be $$ \langle x, x \rangle = \|x\|^2 $$ for any ##x \in \mathbb{R}^2##. (You got it slightly backwards, gopher_p!) I think you're supposed to look at ##\langle e_1 + e_2, e_1 + e_2 \rangle##.