3
votes
1answer
26 views

$(2,1+\sqrt{-5}), (1-\sqrt{-5},2)$ generate the $\mathbb Z[\sqrt{-5}]$-module $\langle 2,1+\sqrt{-5} \rangle \times \langle 2,1+\sqrt{-5} \rangle$

Ok, boring question here (I guess, at least). Let $R=\mathbb Z[\sqrt{-5}]$. Let $M=\langle 2,1+\sqrt{-5} \rangle$ the $R$-module generated by $2$ and $1+\sqrt{-5}$. I am asked to show that $M \times M ...
0
votes
0answers
25 views

Are matroids really a generalization of independence in vector spaces?

The axioms of matroids are (Wikipedia): A finite matroid $M$ is a pair $(E,\mathcal{I})$, where $E$ is a finite set (called the ground set) and $\mathcal{I}$ is a family of subsets of $E$ (called ...
0
votes
1answer
14 views

Morphism of $k$-algebras between abelian group of $n \times n$ matrices and $m \times m$ matrices

Problem Let $k$ be a field and $f:M_n(k) \to M_m(k)$ be a morphism of $k$-algebras ($n,m \in \mathbb N$). Prove that $n$ divides $m$. I have no idea what to do here, I thought that maybe I should ...
0
votes
2answers
24 views

Linear independence and grammar

Let $A$ be a commutative ring. Normally, the linear independence is a property of a subset or a family elements of an $A$-module. But one often sees statements like "$v$ and $w$ are linearly ...
0
votes
0answers
21 views

quotients and direct sums

Let $H$, $K$, $W$, be submodules of a module $M$. Is it true that $(H \oplus K)/W \cong H/W \oplus K \cong H \oplus K/W$? The first seems to follow from 1st isomorphism theorem on the map $\phi = ...
1
vote
0answers
31 views

Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$.

Let $A\in\operatorname{M}_n(F)$. How to prove the following identity. $$\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}.$$ Here $m_A(x)$ the minimal polynomial of $A$ and $C(A)$ is the ...
1
vote
1answer
43 views

Matrices over PID

Let $R$ be a PID and $A,B\in\operatorname{M}_n(R)$ are $n\times n$ matrices such that $\det(A)\sim\det(B)\neq0$,i.e., the ideals generated by $\det(A)$ and $\det(B)$ are the same, does there exist ...
1
vote
2answers
43 views

Show that the homomorphism $\lambda: k[X] \to End_k(V) : p \mapsto p(A)$ corresponding to the $k[X]$-module strucutre of $V$ has a nontrivial kernel.

$\DeclareMathOperator{\End}{End}$ I'm trying to show that: Show that the homomorphism $\lambda: k[X] \to \End_k(V) : p \mapsto p(A)$ corresponding to the $k[X]$-module strucutre of $V$ as in (see ...
5
votes
2answers
77 views

Winning strategies in multidimensional tic-tac-toe

This question is a result of having too much free time years ago during military service. One of the many pastimes was playing tic-tac-toe in varying grid sizes and dimensions, and it lead me to a ...
2
votes
1answer
32 views

Infinite dimensional Vector Spaces and Bases of Quotients

Given a basis $\mathcal{B}$ of an infinite dimensional vector space $V$, will a quotient of $V$ by a subspace $I$ necessarily have a basis? Is there a way of obtaining it using $\mathcal{B}$? My ...
1
vote
1answer
33 views

Finding a suitable basis for a vector space

Let $k$ be a division ring and $V$ be a left $k$-vector space of infinite dimension. Let $E=\mathrm{End}(V)$, defined as a ring of right operators on $V$. My questions are: (1) Why $V$ is a simple ...
1
vote
1answer
55 views

Question concerning a list sorting problem

I have the following question: Let $a,b,c,d$ be four natural numbers with $a \leq b$ and $c\leq d$. I have written a program that produces a list, which has as entries all 2-tuples $(x,y)$ with ...
1
vote
1answer
24 views

Inclusion induce an isomorphism

I have a small question If $E,F$ are two sub-modules of a module $G$, how to prove that the inclusion $E \rightarrow E+F$ induces an isomorphism $(E/E \cap F) \simeq (E + F) / F$ Please Thank you ...
2
votes
1answer
133 views

Is there vector space $\mathbb R $ over $\mathbb R $ with dimension $m\neq1$?

I am searching a new scalar product that with this scalar product the vector space $\mathbb R $over $\mathbb R $ with ordinary vector sum has dimension $m$ , which $m$ is any arbitrary natural ...
1
vote
1answer
46 views

Question about Universal Mapping Property

Now I seem to understand the construction of the tensor product of $R$-Modules by defining $$F_R(M \times N) := \bigoplus_{(m,n) \in M \times N} R\delta_{(m,n)}$$ and constructing the submodule $D$ in ...
2
votes
1answer
57 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
83 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
25 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
18 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
49 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
77 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
85 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
41 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
65 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
169 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
61 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
30 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
42 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
54 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
138 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
66 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
49 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
98 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
59 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
170 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
36 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
144 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
77 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
21 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
38 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
173 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
160 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
43 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
78 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
51 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 $V/U$ ...