Recent content by Lelouch

What I've done is used Fermat's Little Theorem like this \begin{equation*} \begin{split} n^{11a + 21b + 31c} & \equiv n^{11a}n^{21b}n^{31c} \text{ (mod 11)} \\ & \equiv (n^{a})^{11}(n^{b})^{11} (n^{b})^{10}(n^{c})^{11}(n^{c})^{11}(n^{c})^{9} \text{ (mod 11)} \\ & \equiv n^{a}n^{b}...

I don't quite understand. I am really lost on this one. If I am supposed to find a subgroup U with ##45 | |U|##, then this subgroup must have ##|U| = 45, 90, 135, ... ##

Homework Statement Let ##a, b, c \in \mathbb{N \setminus \{0 \}}##. Show that for all ##n \in \mathbb{Z}## we have $$n^{11a + 21b + 31c} \equiv n^{a + b + c} \quad (mod \text{ } 11).$$ Homework EquationsThe Attempt at a Solution We have to show that ##11 | (n^{11a + 21b + 31c} - n^{a + b +...

Homework Statement Decide all abelian groups of order 675. Find an element of order 45 in each one of the groups, if it exists. Homework Equations /propositions/definitions[/B] Fundamental Theorem of Finite Abelian Groups Lagrange's Theorem and its corollaries (not sure if helpful for this...

Thank you. I was able to solve the problem given your explanation of the terminology.

I wanted to ask why the norm was defined as the inner product, but now right before I answered I noticed you added the ##\sqrt{}##. It seems simple I just calculate the respective inner products using the integral, then square root and check whether the inequality holds for the numerical...

Homework Statement Homework Equations I am not sure. I have not seen the triangle inequality for inner products, nor the Cauchy-Schwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm...

I apologize for this. It was like 3 am for me when I wrote this thread and I was tired and was banging my head against a wall with this problem. I was re-doing the theory, the examples in the book by hand and proofs and was mad that I couldn't see where the mistake was. I was even more mad when...

Oh my... I just used the following instead of ##PTranspose## It seems correct, since the eigenvalues are on the main diagonal. However, why can't I define/use the name PTranspose to equal the transpose of P?

I just did that and it gives the following result:

A problem that I have to solve for my Linear Algebra course is the following We are supposed to use Mathematica. What I have done is that I first checked that A is symmetric, i.e. that ##A = A^T##. Which is obvious. Next I computed the eigenvalues for A. The characteristic polynomial is...

I quote from the preface of Tao's Analysis I book and it continues Obviously, the targeted audience are students at the undergraduate level who have not learned real analysis in a rigorous way and not people like you who want to "just look up" something. Hence, it is supposed to be...

Yes this seems to me a better approach. At least for people who get stuck and have to read the proof. I.e. being guided by the author. But he is not saying "spoiler alert I am only interested in this epsilon" he says "let ##\epsilon = \frac {d(l, l')} 3##" and you the reader figure out what...

Yes, if you add variable notation, limit notation and so on. Then, I would. And I would still attempt to prove the proposition on my own. But in the case where I would have no clue what to do, it would benefit.

I actually wanted to write about naming variables in programming languages are also often self-explanatory (if the programmer chose good names not such as int x instead of int score) not like in mathematics. But I do not really have problems with variable naming in mathematics because a...