Recent content by TMO

Taking ## (M + I) = 0 ## and transforming it into RREF gives ##\begin{align}\begin{pmatrix} 1& -1& 0 \\ 0& 0& 0 \\ 0& 0& 0 \end{pmatrix} \end{align}##. The non-pivot columns are two, so the eigenspace is given by ##\begin{align} v_1 - v_2 =& 0& \\ v_2 =& r \\ v_3 =& s \end{align}##...

I need to find the change of basis matrix using eigenvectors and generalized eigenvectors. Why? Because for larger matrices I may not be able to get a nice number partition that allows me to guess the JCF. I know there's a way to do this. How do I do this?

Let ## \begin{align}M =\begin{pmatrix} 2& -3& 0 \\ 3& -4& 0 \\ -2& 2& 1 \end{pmatrix} \end{align}. ## Here is how I think the JCF is found. STEP 1: Find the characteristic polynomial It's ## \chi(\lambda) = (\lambda + 1)^3 ## STEP 2: Make an AMGM table and write an integer partition...

To see the steps I have completed so far, https://math.stackexchange.com/q/3168898/261956 I think there are at least three more steps. The next step is finding the eigenvectors together with the generalized eigenvectors of each eigenvalue. Then we use this to construct the transition matrix...

This is a very obvious question, but I am having trouble concentrating. Let ax ≡ b mod c and let gcd(a, c) | b. How do I convert this equation into Bezout's identity so that I can use the extended Euclidean algorithm?

For any metric ##(X, \rho)## and points therein, prove that ##|\rho(x, z) - \rho(y, u)| \leq \rho(x, y) + \rho(z, u)##. I know that this will involve iterated applications of the triangle inequality...but I still need another hint on how to proceed.

http://web.williams.edu/Mathematics/sjmiller/public_html/302/coursenotes/Trapper_MethodsContourIntegrals.pdf See Type 5 Integrals. I don't understand why J is equal to the original real integral multiplied by a factor of ##2\pi i##. I think the ##2\pi i## comes from the fact that as you go...

Given a ring R, how exactly do I interpret it as a module? A lot of my homework assignments involve treating a ring as "a module over itself" and I don't know precisely what that means.

I think there is a theorem that for a principle ideal domain X, X is divisible iff. X is injective. Tell me, is it iff. or if? It isn't obvious to me. Could you explain why?

Let's suppose that I have an element ##e## of order ##p## in the group of complex numbers whose elements all have order ##p^n## for some ##n\in\mathbb{N}## (henceforth called ##K##), and the module generated by ##(e)## is irreducible. How do I show that the injective hull of the module...

Problem. Let ##p## be a prime integer. Let ##Z_p^\infty## be the set of complex numbers having order ##p^n## for some ##n \in \mathbb{N}##, regarded as an abelian group under multiplication. Show that ##Z_p^\infty## has an unique simple submodule. Attempted solution. The collection of all...

First of all, I cannot use the fact that Q is the cokernel of the map because I do not know how he is even defining k to begin with. Are you allowed to take two objects in a diagram and say "let <foo> be a morphism between <ma> and <mi>" without giving evidence to suggest such a morphism should...

I understand how he concluded that the morphism h existed (An R-module E is injective iff. for all R-module homomorphisms ϕ : M → N and ψ : M → E where ϕ is injective, there exists an R-linear homomorphism θ : N → E such that θ ◦ ϕ = ψ). But how exactly did he conclude that the morphism k...

Given: the short exact sequences 0 → M → E → K → 0 and 0 → M → E' → K' → 0 where M is a left R-module and E and E' are injective left R-modules. Prove: E ⊕ K' ≅ E' ⊕ K. First, let f be the morphism represented by M → E and g be the morphism represented by M → E'. Therefore we can construct a...

Then I don't what to go to grad school. I'll find another use for my time.