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Questions tagged [direct-sum]

For questions about taking the direct sum of groups and other algebraic structures.

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Find subspace $T$ of $\mathbb{R}^{3}$ such that $\mathbb{R}^{3} =V \oplus T$

\begin{array}{l} V=\{( a+2b,2a+8b+2c,a+10b+4c) \ |\ a,b,c\in \mathbb{R}\}\\ =\{a( 1,2,1) +b( 2,8,10) +c( 0,2,4) \ |\ a,b,c\in \mathbb{R}\}\\ =Sp\{( 1,2,1) ,( 2,8,10) ,( 0,2,4)\} \end{array} Then I ...
Shai's user avatar
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Why Do Additive Categories Need Zero Objects? - Motivation

At the moment, I'm trying to develop intuition behind the derived construction of what is an additive category from the (standard) category definition. (i) It seems natural that a category with the ...
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Finite Direct Sum in relation to Finite Direct Product - Additive Category

The question is have is admittedly very basic, but I struggle to find posts (on StackExchange) or proofs addressing it. Let $A$ denote some arbitrary additive category. Hence, the direct sum $(X_1\...
JAG131's user avatar
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Triangle inequality on a direct sum of Lebesgue spaces

I think a priori it is known that for $1 \leq p, q < \infty$ we have that $||(\hspace{0.1cm},)||: L^p(E) \times L^q(E) \rightarrow \mathbb{R}_{\geq 0} $, defined as $||(f, g)|| = (||f||^2_{L^p} + |...
bellumthirio's user avatar
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Matrix representation of direct sum of linear operators

Given two vector spaces $V,W$ over $K$, let $F\in L(V), T\in L(W). $ If $B,U$ are ordered basis for $V,W,$ respectively, show that the matrix $\left[F\right]_B\bigoplus \left[T\right]_U$ represents ...
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Can I derive vector space-equivalent from direct sum?

Let $V$ be a vector space and $A_1, A_2, B_1, B_2$ are subspace of $V$ such that $$A_1 \leq B_1, \ A_2 \leq B_2,\ A_1\oplus A_2 = B_1 \oplus B_2$$ Then $A_1 = B_1 , A_2 = B_2$? What I found : Due to ...
TheoryMongus's user avatar
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Show that two summands of a direct sum are invariant under a linear operator

Given two linear operators $T,S\in L(V)$ such that $T$ and $S$ commute ($T\circ S=S\circ T$) and $T^2=T$. Show that: (a) $V=ker(T)\bigoplus im(T)$. (b) each summand is invariant under $S$. So far I ...
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Is a finitely generated module over a hereditary ring always finitely presented? When does $Ext^{n}( M, -)$, for $n \geq 0$, commute with direct sum?

In the §6 Appendix II (2) of the article Gorenstein projective modules says that: Lemma 1 : Let $R$ be a ring. If $M$ is a finitely generated $R$-module, then $$Ext^{n}( M, \oplus_{i\in I}\ N_{i}) \...
Liang Chen's user avatar
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External direct sum $U_1\oplus U_2$ isomorphic to $U_1+U_2$ does not necessarily imply that $U_1\cap U_2 = \{0\}$ [closed]

Let $V$ be a vector space (not necessarily finite-dimensional) and let $U_1,U_2\subset V$ be subspaces. If $U_1\cap U_2=\{0\}$, then the surjective linear map $\phi\colon U_1\oplus U_2 \to U_1+U_2$ ...
Apollo13's user avatar
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4 votes
1 answer
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Irreducible but not absolutely irreducible representations

Let $\mathbb F_q$ be a field of $q$ elements where $q$ is an odd prime power. Let $G$ be a finitely generated group and $\rho:G \to \operatorname{GL}_2(\mathbb F_q)$ be an irreducible representation ...
Conjecture's user avatar
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The number of direct sum of elementary abelian 2-groups

Let $G=(Z_2)^n$, I want to know the number of direct sum of $G$($G=H \oplus K$) or a fine upper bound. For $G=Z_2 \oplus Z_2$, I have calculated that all of its direct sum decomposition is as follows: ...
zeyu hao's user avatar
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$n$th symmetric power of a superspace

Given a vector space $V$, we consider the (trivial) associated even superspace $V\oplus 0$ and odd superspace $0\oplus V$. For any (super) vector space $W$ we define the $n$th symmetric power as $$ \...
Albert's user avatar
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Direct sum of orthogonal complements

Prove that if $$U \oplus W = V,$$ then $$U^ \perp \oplus W^ \perp = V,$$ where $U^ \perp$ is an orthogonal complement of $U$. I've tried to use the fact that $$U \oplus U^ \perp = V, \ W \oplus W^ \...
Gleb Cloudy's user avatar
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Generating a free module

Let $R$ be a ring with $1$ and let $M$ be an $R$-module. Let $A \subseteq M$. Is it true that the following are equivilent. $A$ is linearly independent. $R \cong Ra$ and $$\sum_{a \in A}Ra = \...
Ethan Kharitonov's user avatar
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The projection of the polynomial onto U parallel to V

Prove that the space $P_3$ of polynomials of degree no higher than $3$ is the direct sum of subspaces $U$ and $V$ and find the projection of the polynomial $t^3$ onto $U$ parallel to $V$, where \begin{...
Alice P.'s user avatar
4 votes
1 answer
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Isomorphisms from the external sum of modules to the internal sum

Is this true: Let $R$ be a ring with a $1$. Let $(N_{i})_{i \in I}$ be a collection of $R$-submodules of the $R$-module $M$. If $$\sum_{i \in I}N_{i} \cong \bigoplus_{i \in I}N_{i}$$ then the map $f: \...
Ethan Kharitonov's user avatar
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1 answer
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A problem about diagonalize invariant subspaces [closed]

Let $V$ be a non-zero finite-dimensional vector space, A belongs to End($V$). Also, for any invariant subspace $M$ of A, there exists an invariant subspace $N$ of A such that $V=M\oplus N$. Prove: A ...
淘宝者's user avatar
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If $X=M\oplus N$, is it true $(A\cap M)\oplus (A\cap N)\subseteq A$?

Let $X$ be a norm space and $X=M\oplus N$ where $M,N$ are closed subspace of $X$. Take a closed set $A\subseteq X$. I know that $(A\cap M)\oplus (A\cap N)\neq A$, for example consider $X=\mathbb{R}^2$,...
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Generating sets of an Abelian group decomposed into a direct sum

I’m not sure about the correctness of my assumption about the generating sets of decomposable Abelian groups, so I will be very grateful if you can tell me whether I’m right or wrong, and if I’m wrong,...
moonruleni9ne's user avatar
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Proving existence and uniqueness of direct sum of representations of unital $C^*$-algebras

From Sunder's Functional Analysis: Spectral Theory, Exercise 3.4.3. Let $\{ \pi_i : A \to \mathcal{L}(\mathcal{H}_i) \}_{i\in I}$ be an arbitrary family of representations of (an arbitrary unital $C^*...
user avatar
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2 answers
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About the direct sum.

Today I decide to study some topics of Algebra and then faced up with the definition of the direct sum of modules. To let us on the same page, the definition that I'm talking about is Given a ring $R$...
Paulo Estêvão's user avatar
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35 views

Decomposition of $K$-finite functions

It is a theorem that a representation $(\pi,V_\pi)$ of a compact group $K$ decomposes as a direct sum of irreducible representations. My question is about Deitmar's treatment (Automorphic Forms page ...
jshpmm's user avatar
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4 votes
1 answer
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Recover direct summands in derived category?

Let $E,F\in D^b(X)$, where $D^b(X)$ denotes the derived category of coherent sheaves on some smooth variety $X$. I am thinking about the following question: If $E \oplus E[1] \simeq F \oplus F[1]$, ...
Yuri's user avatar
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Dimension of tensor product vs. dimension of direct product

I am really sorry if this is trivial but I come from Physics and it is really confusing for me to understand what is going on and looking at the answers on both Physics and Math SE sent me through a ...
QFTheorist's user avatar
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1 answer
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Find a subspace $W$ of $P_{2}(\mathbb{R})$ such that $P_{2}(\mathbb{R})$ = $ U \oplus W$

Let $P_{2}(\mathbb{R})$ be the real vector space of all real polynomials of degree at most 2, equipped with the usual addition and scalar multiplication operations, and ܷ$U$ = { $p(x)\in P_{2}(\mathbb{...
Felix's user avatar
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Exercise 9, Section 4.2 of Hungerford’s Algebra

If $F_1$ and $F_2$ are free modules over a ring with the invariant dimension property, then $\text{rank} (F_1 \oplus F_2) = \text{rank} F_1 + \text{rank} F_2$. I have written proof of this exercise ...
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Exercise 17, Section 4.1 of Hungerford’s Algebra

(a) If $R$ has an identity and $A$ is an $R$-module, then there are submodules $B$ and $C$ of $A$ such that $B$ is unitary, $RC= 0$ and $A=B\oplus C$. [Hint: let $B=\{1_Ra\mid a\in A\}$ and $C=\{a\in ...
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Give a non-zero $R$-module $M$ such that $M \oplus M \cong M$ [duplicate]

For some reason this hasn't been asked before and I can't seem to find an example. Here $R$ is a commutative ring. Find a non-zero $R$-module $M$ such that $M \oplus M \cong M$
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How are "types" defined in this module?

Note: This might end up being a question about a simple concept that I forgot about (I am very tired at the time of writing this, after all), so maybe try skipping to the bottom. I'm learning about ...
iwjueph94rgytbhr's user avatar
1 vote
1 answer
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Trouble understanding direct sum in smith normal form proof

I'm struggling to understand this step in this proof in Rotman's advanced modern algebra. The related theorems are: Let R be a euclidean ring, let F be a finitely generated free R-module, and let S ...
ctk's user avatar
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Direct sum of Hilbert/Sobolev Space?

Our professor the other day wrote the following on the blackboard: "Let $u(x)=(v(x),w(x))$ be such that $u \in H^{2,1}$ where $H^{2,1}=H^2 \oplus H^1$." He then mumbled something about this ...
kaithkolesidou's user avatar
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Let $V$ be a finite dimensional vector space and $W$ be a subspace

Let $V$ be a Finite Dimensional Vector Space and $W$ be a non trivial proper subspace of $V$ then the linear span $ \langle V- W\rangle $ =$V$? The statement is true, & the issue here I'm facing ...
Gajjze's user avatar
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5 votes
1 answer
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How are submodules characterized with respect to the isomorphism of the direct sum of multiple modules?

Consider two rings $R_1$ and $R_2$ each with their own identities, and their direct product $R_1 \times R_2$. It is well-known that the ideals of $R_1 \times R_2$ are all of the form $I_1 \times I_2$, ...
Liang Chen's user avatar
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Necessary and sufficient condition for a $ Z \oplus Z$ base

Prove that the elements $b_1=(x_1,y_1)$, $b_2=(x_2,y_2)$ form a base of $Z \oplus Z$ if and only if, $ x_1y_2-y_1x_2=±1 $. This exercise is from ANEIS E MODULOS ,Francisco Cesar Polcino Milles. ...
Renato lorentz's user avatar
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$\big\langle B_1\cup\cdots\cup B_p\big\rangle=\langle B_1\rangle\oplus\cdots\oplus\langle B_p\rangle$

Let \begin{align} B_1\,&=\,\big(u_{11},\dots,u_{1n_1} \big),\dots,B_p\,=\,\big(u_{p1},\dots,u_{pn_p} \big) \end{align} be disjoint linearly independent sets such that $$B\,=\,B_1\cup\cdots\cup ...
PermQi's user avatar
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Proving two modules are free based on their direct sum [duplicate]

I have given two modules $M$ and $N$ over a local ring $R$. I also know that $M \oplus N \cong R^n$ for some $n\in \mathbb{N}$. I then have to prove that both $M$ and $N$ are free modules. Since $M \...
MarlonButBetter's user avatar
1 vote
1 answer
80 views

Isomorphic to external direct sum but internal direct sum not defined

Previously I asked a questionon whether isomorphic to external direct sum implies "is internal direct sum". The answer turns out to be no, but whenever the internal direct sum is "...
wsz_fantasy's user avatar
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2 votes
1 answer
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Is the direct sum $M\dot{+} N$ closed in $\ell^\infty (\mathbb{N})$?

Given two subspaces $M= \{ a \in \ell^\infty (\mathbb{N}) | a_{2n}=0 \}$ and $N= \{b \in \ell^\infty (\mathbb{N}) | b_{2n-1}=nb_{2n} \} $, is the direct sum $M\dot{+} N$ a closed subspace of $\ell^\...
mikasa's user avatar
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1 vote
1 answer
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Direct sum question problem

Find $k \in \mathbb{R}$ such that $\mathbb{R}^3=S \oplus T$, with $S=\operatorname{span}\left\{\left[\begin{array}{c}-3 \\ 4 \\ 1\end{array}\right]\right\}$ and $T=\left\{(x, y, z) \in \mathbb{R}^3: k ...
Rajashi's user avatar
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2 votes
1 answer
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How to conclude that this is a semisimple algebra

Let $k$ be a field, $R$ a $k$-algebra and $J$ an ideal of $J$. I have proved that $$R/J\cong k \oplus \ldots \oplus k$$ which means that $R/J$ is semisimple as a $k$-module. But I want to conclude ...
kubo's user avatar
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Exponent of $\Bbb Z$ in tensor product

In computing $$ \newcommand{\Z}{\mathbb{Z}} \newcommand{\op}{\oplus} \Bigl( \Z^2 \op (\Z/6\Z) \op (\Z/126\Z) \Bigr) \otimes_\Z \Bigl( \Z \op (\Z/45\Z) \op (\Z/495\Z) \Bigr), $$ do I get as exponent ...
darkside's user avatar
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1 vote
1 answer
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$\mathrm{Hom}_A(\prod_i M_i, N) \cong \prod_i \mathrm{Hom}_A(M_i, N)$ and $\mathrm{Hom}_A (M, \bigoplus_i N_i) \cong \prod_i \mathrm{Hom}_A(M, N_i)$?

I am reading the notes of Pierre Shapira on Algebra and Geometry. He writes the following: Let $A$ be a $k$-algebra, $M \in Mod(A)$. Then $\mathrm{Hom}_A(M, \prod_i N_i) \cong \prod_i \mathrm{Hom}_A(M,...
liv's user avatar
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1 vote
1 answer
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Presentation of direct sum

I find that presentations are the most useful tool to get an intuitive feeling about the nature of some group construction. I have read here at mathstack that given $\{G_i = \langle X_i \mid R_i \...
Lucas Giraldi's user avatar
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Module written as direct sum

I'm taking Non Commutative Algebra course and today my professor left the following exercise: if $R$ is a ring and $M$ is an $R$-module writtem as $$M = \sum_{i=1}^{n} V_i,$$ where each $V_i$ is an ...
Guilherme Costa's user avatar
1 vote
1 answer
39 views

Is the projections to a factor in a submodule of a direct sum over a unital ring, a submodule of the ring?

Let $R$ be a ring with unity (not neccessarily commutative), and say we have a finite direct sum $$\bigoplus_{i = 1}^{n}R$$ and a submodule $$K \subseteq \bigoplus_{i = 1}^{n} R.$$ If we now view $R$ ...
Ben123's user avatar
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1 vote
1 answer
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Show that $(Ker(\psi))^{\perp} \subset Im(\phi)$

I'm trying to solve the following problem: Let $V$ and $W$ be two finite dimensional vector spaces over $\mathbb{C}$ with inner products $p$ and $q$ respectively. Let $\psi: V \rightarrow W$ be a ...
MC2's user avatar
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1 vote
1 answer
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Pontlyagin dual of direct sum,$ \widehat{\bigoplus_{i\in \Lambda} M_i} = \prod_{i\in \Lambda} \hat{M_i}. $

Let $M_i$ (for $i \in \Lambda$) be a family of abelian groups. Let $\bigoplus_{i\in \Lambda} M_i$ denote the infinite direct sum of the groups $M_i$. Let $\hat{M}$ denote the Pontryagin dual of the ...
Pont's user avatar
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1 vote
1 answer
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Codimension of a linear operator restricted to a finite dimensional subspace

Suppose that $K$ is a compact linear operator acting on a Banach space $X$ into itself. Then there exist closed subspaces $N$ and $Z$ with $N$ finite dimensional, $Z \subset \ker{K}$ and $$ X = N \...
liamsi Meean's user avatar
1 vote
1 answer
87 views

submodule of completely reducible module is completely reducible

I attempted to solve problems from the textbook 'Basic Abstract Algebra' by P.B. Bhattacharya, S.K. Jain, and S.R. Nagpaul, specifically Chapter 14, Section 4, but encountered difficulties in ...
N00BMaster's user avatar
2 votes
0 answers
51 views

the non-abelian subgroups of the Lamplighter group

The lamplighter group can be defined by the semidirect product: $$ L_2=(\mathbb{Z} _2) \wr \mathbb{Z} \cong \bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_{2} \rtimes_\phi\mathbb{Z},$$ where $\phi(1)$ &...
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