2
votes
1answer
56 views

Linear dual of vector fields

Suppose that $M$ is a smooth manifold and $\mathfrak{X}(M)$ is the set of smooth vector fields on $M$. There are basically two different linear structures on $\mathfrak{X}(M)$: 1.) $\mathfrak{X}(M)$ ...
1
vote
1answer
68 views

Why is the Ideal Norm Multiplicative?

I had asked this question before and got a partial answer. Let $A$ be a Dedekind domain with quotient field $K$, $L$ a finite separable extension of $K$ of degree $n$, and $B$ the integral closure of ...
0
votes
0answers
23 views

Smith normal forms and a math program

I am interested to know the Smith normal form of $4 \times 2$ matrices $M$: The two cases of my interests are: (1). $$M_1= \begin{pmatrix} 3 & 0\\ -5 & 4\\ 4 & -5\\ 0 & 3 ...
0
votes
1answer
13 views

$x\in{\rm span}(S\cup\{y\}), x\notin{\rm span}(S)$ implies $y\in{\rm span}(S\cup\{x\})$

If $x\in{\rm span}(S\cup\{y\})$ and $x\notin{\rm span}(S)$, then $y\in{\rm span}(S\cup\{x\})$ This statement is simple enough if ${\rm span}$ is defined in terms of finite linear combinations: if ...
1
vote
1answer
35 views

The Chinese remainder theorem for modules.

Let $M$ be a free $\mathbb{Z}$-module. Let $a_1,a_2\in \mathbb{N}$ such that $\gcd(a_1,a_2)=1$. Then $${M}/{a_1a_2M}\cong {a_1M}/{a_1a_2M} \oplus {a_2M}/{a_1a_2M}.$$ Is the above statement true? ...
0
votes
0answers
11 views

Verification of step in a proof of the decomposition of primary f.g torsion modules over PIDs?

I was reading about the decomposition of finitely generated primary torsion modules over PIDs, and though of an alternative way to do the "inductive" step. Since it is substantially simpler than the ...
-1
votes
1answer
58 views

Basic short exact sequence exercise in $\mathbb{Z}$-modules [closed]

I was wondering how to properly argue the proof of the following: a) Prove that there is an exact sequence $0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}_n \to 0$ (consider $f : \mathbb{Z} \to ...
2
votes
2answers
73 views

How to find a basis for a tricky 2x 2 matrices vector space

Consider the vector space of 2 x 2 matrices :\begin{bmatrix}a&b\\0&c\end{bmatrix} such that a and c are rational numbers and b is a real number with rational numbers as the field of this ...
0
votes
0answers
39 views

How do i show that something has the structure of a vector space and how do i prove something to be linearly independent?

Im not sure what to do.....where do i begin? $\quad$ Throughout, $V$ denotes a vector space of finite dimension $n$ over the field $F$, and $\tau$ an element of ${\cal L}(V).$ Recall the structure of ...
2
votes
1answer
62 views

Structure of a module and finite fields

I am trying the following problem which appeared on an exam for a grad course: Determine the structure of finite field $F$ as a module over $F_p[x]$ when $xv=Lv$. Here $L(z)=z^p$ and ...
3
votes
2answers
120 views

Every vector space has a basis using minimal spanning set.

We have seen the argument for proving the above statment using Zorn's Lemma by asserting the existence of a maximal linearly independent set which serves a basis. In finite dimensional vector space, a ...
1
vote
2answers
59 views

Can a the field of fractions or quotient field F of an integral domain R be free over some set as an R-module?

We know any integral domain R when extended to a quotient field F, then F is free as an F-module on the set {1}. Can this field be free over some set as an R-module.
1
vote
0answers
26 views

Composition series of finite length modules

Given a field $K$ of characteristic zero. Let $V$ be a finite dimensional $K$-vector space and let ${f} \in End_K(V)$. Then $V$ can be regarded as a $K[X]$-module by the action: ...
2
votes
0answers
85 views

Help in this proof in Lang's Algebra book (really elementary doubt)

My doubt is really elementary: Let $(f_1,\ldots,f_n)$ be a row matrix with some component with leading coefficient $1$ and $d$ the smallest degree of a component of $f$ with leading coefficient $1$. ...
0
votes
0answers
23 views

Differential operators in the language of modules

I am reading articles about differential operators. Authors try to treat differential systems , by studying the differential operator on a vector space in the language of modules. In this manner ...
2
votes
1answer
37 views

Tensor product of certain algebras

If S is an $R$-algebra of finite dimension, and $A$ is an algebra of infinite dimension, then is $S \otimes S \otimes A \cong S \otimes A$?
4
votes
0answers
52 views

Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
0
votes
0answers
23 views

Linear Differential Equation with Quasiperiodic Coefficient: A Question

I am reading the book Synergetics. The following passage is from the book: Assume that in the differential equation $\dot{q}= a(t)q$, where $a(t)$ is quasiperiodic, i. e., we assume that $a(t)$ ...
5
votes
1answer
129 views

Artin exercise on free modules homomorphism

2.3 Let $A$ be the matrix of a homomorphism $\varphi:\mathbb Z^n\to\mathbb Z^m$ of free $\mathbb Z$-modules. (a) Prove that $\varphi$ is injective if and only if the rank of $A$, as a real ...
1
vote
2answers
62 views

About definition of “direct sum of $p$-vector subspaces”

In the books 1 and 2, in Somme directe d'une famille de sous-espaces vectoriels, I am reading the following: 1) let $E,F$ two vector subspaces of $V$, $E+F$ is direct sum, $E+F \doteq E\oplus F$, if ...
2
votes
1answer
41 views

direct products and direct sums of skew fields

If F is a skew field then are the arbitrary direct sums of F-modules isomorphic to their direct products (over the same index). I mean, if R is a division ring, and $\{M_i\}_{i\in I}$ some familly ...
2
votes
1answer
46 views

Computing the kernel of a homomorphism from a free $\mathbb{Z}$-module

Given a finitely generated free $\mathbb{Z}$-module $N$ and a homomorphism $\varphi:N\to M$, is there an easy way to compute the kernel of $\varphi$? Since the kernel is also a free ...
2
votes
1answer
75 views

Can we classify commuting pairs of matrices up to conjugacy?

Recall that two $n\times n$ matrices over $\mathbb{C}$ are conjugate if and only if they have the same Jordan canonical form. Question. Is there a similar classification for commuting pairs of ...
3
votes
1answer
55 views

Determinant of a Linear Transformation is Unique Up to a Unit

Can anyone explain the following assertion from a textbook I'm reading? "...$R$ is a Dedekind domain, $K$ its quotient field, $U$ a finite dimensional vector space over $K$ of dimension $n > 0$ ...
5
votes
1answer
165 views

Generalization of Cayley-Hamilton

I'm having trouble following a proof of this generalization of the Cayley-Hamilton theorem: Suppose that $M$ is an $A$-module generated by $n$ elements, and that $\varphi \in ...
1
vote
1answer
34 views

Duals of modules over algebras

Suppose $A$ is an associative and commutative $\mathbb{R}$-algebra with unit $1_A$ and $M$ is an $A$-module. Let's write $\cdot_A:A\times M\to M$ for the $A$-scalar multiplication on $M$. 1.) It ...
0
votes
2answers
137 views

What's a Dual Basis?

If $L/K$ is a finite extension of fields with $v_1, ... , v_n$ a basis, then we have an isomorphism of $K$-modules $L/K \rightarrow Hom_K(L,K)$, where a basis for $Hom_K(L,K)$ is the "dual basis" of ...
1
vote
1answer
72 views

Invariant Factor Decomposition of Integer Matrices

Suppose $M$ is an $n \times n$ matrix with integer entries. If we think of $M$ as acting on $V=\mathbb{Q}^n$, then we can think of $V$ as a $\mathbb{Q}[M]$-module and have that $V$ is isomorphic to a ...
0
votes
0answers
20 views

System of linear equations with modulo elements

Transform the following $3 \times 3$ matrix $A$ with entries in $\mathbb{Z}/5\mathbb{Z}$ into reduced row echelon form (with $[1]$ as Pivot element and $[0]$ above the Pivot element). State out the ...
0
votes
1answer
37 views

field extension $F(x)=F(x^2)$ [duplicate]

Let $x$ be algebraic over $F$ such that the field extension $F(x):F$ satisfies $[F(x):F]$ odd. Then prove $[F(x):F]=[F(x^2):F]$ hence $F(x)=F(x^2)$. How to prove? I only obtained the proof for the ...
4
votes
3answers
156 views

similar matrices over $\mathbb{Z}$

Let $A,B$ be $2\times 2$ matrices over $\mathbb{Z}$. Suppose $x^2+x+1$ is the characteristic polynomial for both $A,B$. Determine whether $A,B$ are similar to each other over $\mathbb{Z}$. How to ...
0
votes
1answer
124 views

matrices with same minimal and characteristic polynomials are conjugate

Let $A$, $B$ be two $n\times n$ matrices over $\mathbb{C}$. If $A$, $B$ have the same minimal polynomials and characteristic polynomials, then can we prove that $A$, $B$ are conjugate? i.e., does ...
1
vote
0answers
39 views

Vanishing criterion of pure wedges

Let $R$ be a commutative ring, $M$ some $R$-module, and $m,n \in M$. Is there some criterion when $m \wedge n = 0$ in $\Lambda^2(M)$? There are some sufficient criterions, for example that $m \in ...
3
votes
1answer
73 views

Proof of Clifford's theorem for modules

http://en.wikipedia.org/wiki/Clifford_theory#Proof_of_Clifford.27s_theorem I've a very easy question that I just can't seem to find the answer to. I'm self-studying so I can't ask anyone else. ...
1
vote
1answer
50 views

Any submodule U of V such that the module V/U is completely reducible must contain the radical?

Want to prove: Given an $R$-Module $V$ and $rad(V)$ which I define to be the intersection of all maximal submodules of $V$. I want to show that if, for some submodule $U$ of $V$, we have $V/U$ ...
1
vote
1answer
20 views

Tensor over division rings

Let $D$ be a division ring containing a field $k$ in its center such that $D$ is finite dimensional over $k$. Consider the left $D$-module $D \otimes_{k} V$ where $V$ is a finite dimensional k-vector ...
0
votes
1answer
57 views

Prove equivalence: Internal Direct Sum of Modules (HW)

Let $M$ be a module and $M_1, M_2, M_3,..., M_n$ be submodules of $M$ verify that the following are equivalent: (1) $M$ is the internal direct sum of the $M_i$'s. (i.e. ...
1
vote
1answer
64 views

Homework basic abstract algebra

My question is as follows: $R$ is a ring such that for all $x \in R, x^2=x$ $p$ is a prime ideal of $R$. Show that $R/p$ (R modulu p) has exactly 2 elements. What I did: $x^2=x$ $x^2-x=0$ ...
1
vote
0answers
79 views

Irreducible modules - semisimple algebras and endomorphism rings

Let $A$ be a finite dimensional, semi-simple $k$-algebra and $V$ and irreducible $A$-module. I am trying to prove the following claim: If $B = \text{End}_A(V^{\oplus r})$ then $$W = ...
3
votes
0answers
104 views

R-modules over a vector space

Let $V = \mathbb{C}^3$, $A$ be the matrix with column vectors $e3, e1, e2$ (where $e1, e2, e3$ are the standard basis vectors for $\mathbb{R}^3$). $R = \mathbb{F}[X]$. Let $V_a = V$ be the R module ...
1
vote
0answers
230 views

Spinor representation and Clifford modules

Let $V$ be an even-dimensional real inner product space. We denote the Clifford algebra of $V$ by $C(V)$ and the spinor representation by $S$. For a finite-dimensional $\mathbb Z_2$-graded complex ...
0
votes
1answer
53 views

There's a bijective correspondence between complements of a submodule $N \leqslant M$ and extensions of $\sigma : N \approx M_1$.

Let $N$ be a submodule of $R$-module $M$. let $\sigma : N \approx M_1$ be an $R$-isomorphism. Then the correspondence $H \to \bar{\sigma}$, where $\ker(\bar{\sigma}) = H$ is a bijection from the ...
1
vote
0answers
70 views

A basis of a module is a minimal spanning set.

This is from Roman's Advanced Linear Algebra. Theorem 4.4 Let $\mathcal{B}$ be a basis for an $R$-module $M$. Then $(1) \ \mathcal{B}$ is a minimal spanning set. $(2) \ \mathcal{B}$ is a ...
6
votes
1answer
220 views

Obtain a basis of invertible matrices for $M_n(D)$, where $D$ is an integral domain

Let $D$ be an integral domain. Prove that $M_n(D)$ has a basis consisting of $n^2$ invertible matrices. Consider $M_n(D)$ as a $D$-module. Define invertible elements $V_{nn}=I_n$, $V_{n-1,n-1} = ...
3
votes
1answer
123 views

Similar matrices

Let $A$ be a complex $n×n$ matrix with $tr(A) = 0$. Show that there exists an invertible $n×n$ matrix $B$ such that all diagonal entries of $BAB^{-1}$ are zeros. $\bf Edit:$ Is the fact that $\forall ...
1
vote
1answer
26 views

Is the dual of an equivariant metric equivariant?

Let $g$ a finite dimensional $K$-vector space, and let $g:V \otimes V \to K$ be an inner-product. If As usual, we can use the musical isomorphisms of $g$ to define an inner product on $V^*$, which we ...
3
votes
2answers
86 views

Example of modules over PID's

From Advanced Modern Algebra (Rotman): Definition If $M$ is a finitely generated torsion $R$-module, where $R$ is a PID, and $$M= R/(c_1) \oplus R/(c_2) \oplus ... \oplus R/(c_t),$$ where $t \geq ...
0
votes
1answer
153 views

Infinite tensor product definition

My question is short and popular: how to define an infinite tensor product of modules over a ring? So, there is an infinite set $I$ and $A$-modules $M_i$. I should understand what $\otimes_{i\in I} ...
5
votes
2answers
400 views

How does one obtain the Jordan normal form of a matrix $A$ by studying $XI-A$?

In our lecture notes, there's the following example problem. Find a Jordan normal form matrix that is similar to the following. $$A=\begin{bmatrix}2 & 0 & 0 & 0\\-1 & 1 & 0 ...
3
votes
1answer
70 views

Hamiltonian Quaternions: A Call for Counterexample for ($AB=I_m \implies n≥m$)

How can I build a counterexample to $AB=I_m \implies n≥m$ in the ring of Hamiltonian quaternions? Notice that the vectors $(1,i)$ and $(j,k)=(1,i)j$ are linearly dependent as vectors of the right ...