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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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Sum of subsets is the whole set?

Let $A$ be subset of $\mathbb{Z}_n$. Define $A \oplus A = {\{a \oplus a’ : a \in A, a’ \in A}\}$, where $a \oplus a’ = a+a \mod n$. If $A$ is not contained in a coset of a proper additive ...
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Distributivity of tensor product over a direct sum

Let $\mathcal{H}, \mathcal{K}$ be finite dimensional Hilbert spaces and consider the space $$\left(\mathcal{H} \oplus \mathcal{K}\right) \otimes L^2(\mathbb{R}).$$ I would like a reference to show ...
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Let $A$ be a finite abelian group and $A\cong \oplus_{k=1}^s\mathbb{Z}/p_k^{\alpha_k}$

Let $A$ be a finite abelian group and $A\cong \mathbb{Z}/p_1^{\alpha_1}\mathbb{Z}\oplus\mathbb{Z}/p_2^{\alpha_2}\mathbb{Z}\oplus...\oplus\mathbb{Z}/p_s^{\alpha_s}\mathbb{Z}$, where $p_i$ is prime and $...
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About orthogonal complement in Gilbert Strang's “Linear Algebra and its Applications 2nd Edition”

I am reading Gilbert Strang's "Linear Algebra and its Applications 2nd Edition". On p.137(3.4 The PSEUDOINVERSE AND THE SINGULAR VALUE DECOMPOSITION), he wrote "any vector can be split into two ...
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Is there a nice way to express the quotient of the direct product of a family of groups with the corresponding direct sum?

Let $I$ be an arbitrary (finite or infinite) set of indices and $\{G_i\}_{i\in I}$ a family of groups indexed by $I$. The direct product of the family, denoted by $\prod_{i\in I}G_i$, is the set of ...
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Direct sum of $n$ (infinite) cyclic groups isomorphic to direct sum of $n$ copies of $\mathbb{Z}$?

I'm currently selfstudying some algebra and i am currently covering the various equivalent definitions of free abelian groups. However, in order to understand why these definitions are indeed ...
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Group as direct sum of cyclic groups

What are necessary conditions for a cyclic group $G$ to be a direct sum of cyclic groups? I saw somewhere that $G$ must be a non $p$-group. But I couldn't prove it. Thank you for your hints/help
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Quotient of module versus direct sum

It's maybe stupid question but I want to be sure about what I've learned. Is the quotient $R$-module $R/(p^n) \simeq R/(p) \oplus...\oplus R/(p)$ in general, if not when? I think over $\mathbb{Z}$-...
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Find sum of Subspaces combination of constraints

Let $ S = \langle \{ x^2-1,x^2+1\} \rangle $ and $ T = \{p\in P_2(\mathbb{R}), p(x)= ax^2+bx+c : a+b=0,c=0\} $ $\color{black}{a) \ Find \ S+T}$ From the definition of Sum in subspaces, we take e.g $...
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Decompose vector fields on product manifolds

So, I know that tangent bundle of a product manifold $M \times N$ splits in a sum $$ T_{(x,y)}(M \times N) = T_xM \oplus T_yN, $$ so that is obvious that the sum $X \oplus Y$ of smooth vector fields $...
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Non-Fibonacci numbers, appear from Fibonacci Sums …?

(Please consider that i am not the best on explaining things when it comes to mathematics, so i will try my best.) Lately this month i had an idea, to test if the Sum of Fibonacci Numbers always ...
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The direct sum of two finitely generated algebras is finitely generated

Let $X$ and $Y$ be sets of variables. Let $R$ be a commutative ring. Suppose $S$ and $T$ are finitely generated $R$-algebras. Then $S\cong R[X]/I$ and $T\cong R[Y]/J$ for some $R[X]$-ideal $I$ and $R[...
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Sum dissapearing when we assume some elements to be constant over time

I have the dividend discount model, which is the following expression: $$ P_{j,t} = \sum_{\tau=1}^{\infty}D_\tau(1+g)^\tau(1+r)^{-\tau}=\frac{D_{\tau+1}}{r-g} $$ Where $D_{\tau}$, is the dividend at ...
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Direct sum factorization of polynomials

I have been recently reading the paper "Mixed finite elements for second order elliptic problems in three variables" by Brezzi et. al. I noticed the claim in the proof of Lemma $2.1$, which basically ...
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If $\bigoplus_{i=1}^\infty \mathbb Z \cong Y \oplus \mathbb Z$, is $Y \cong \bigoplus_{i=1}^\infty \mathbb Z$?

Since $\mathbb Q$ is not path-connected, I know that $H_0(\mathbb Q) \cong \bigoplus_{i=1}^\infty \mathbb Z$. Also, $H_0(\mathbb Q) \cong \tilde H_0(\mathbb Q) \oplus \mathbb Z$. So, is it true that ...
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References on the submodules of a direct sum of copies of $\mathbb{Z}$

Let $\mathbb{P}$ be the set of all prime numbers. Consider the $\mathbb{Z}$-Module $\mathbb{Z}^{(\mathbb{P})}$, that is, the external direct sum of copies of the additive abelian group $\mathbb{Z}$ ...
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real world applications of direct sums

I understand how direct sums work and how they can be useful in proving certain conditional statements in linear algebra but it seems to me that direct sums are only useful in abstract settings. I was ...
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Direct Sum and Linear Operators

I'm struggling with a question on linear operators and direct sums. If someone could possibly help me out here that would be great. The question is as follows: Let $V$ be a vector space. Let $...
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If $V = \text{null}(T-\lambda I) \oplus \text{range}(T-\lambda I)$, then $T$ is diagonalizable?

$V$ is a finite-dimensional complex vector space and $T \in L(V)$ ($L(V)$ is the set of all linear maps from $V$ to itself), and $\lambda$ is arbitrary in $\mathbb{C}$. I know $T$ is diagonalizable ...
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linear map between vector spaces/ Direct sum/ Kernel and image

Let $L: V \rightarrow W$ be a linear map between two vector spaces (maybe infinite dimensional). Then is it always true that V = $U \bigoplus (V/U)$, where U is a finite-dimensional subspace of V. (...
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External vs Internal Direct Sum

So we've proven in class that when the sum of two sets, say vector spaces, is such that each element can be uniquely expressed as the sum of two elements in each set, then it is an Internal Direct Sum ...
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Showing two spaces not isometrically isomorphic

Let $X$ be a real Banach space. Consider the direct sum $X\oplus X\oplus X$ with the norm $$\|(x,y,z)\|_1=\|x\|+\|y\|+\|z\|\text{ for all }x,y,z\in X.$$ I want to show that this space is not ...
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A difficulty in understanding the proof of distributivity of tensor products over direct sums for modules.

Here is the proof: But I do not understand the following: 1-why the function needed to be bilinear to use the universal property? 2- what is he doing starting from the paragraph that starts with ...
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Torsion-free Abelian Groups of Finite Rank and Free Groups (Fuchs) - Self study

I want to solve the following problem (Fuchs, "Infinite Abelian Groups", Vol.$2$, pp. $153$ Ex. $4$): "Let $A$ be a torsion-free group of finite rank $n$ and $F$, $F'$ free subgroups of $A$ of rank $...
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Explicit example of a direct sum over two simple groups

I want to understand the notion of a direct sum properly. I learn best via examples so it would very helpful if I could get a concrete example using two simple groups. for examples suppose we want to ...
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37 views

Prove that two subspaces of a vector space intersect only at 0

Let $V \subset \mathbb{R}^n$ & $W \subset \mathbb{R}^m$ with set of basis $S_V=\{v_1,v_2,...,v_n\}$ and $S_W=\{w_1,w_2,...,w_m\}$. The vector space spanned by these basis vectors is the direct sum ...
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Weak Direct Powers

Given any non-empty set $X$, I want to consider the weak direct power $\mathbb{Z}_{n}^{(X)}$. But what is this set defined as? Is it $\mathbb{Z}_{n}^{(X)}=\left \{\mathbf{a}\in \prod_{x\in X}\mathbb{Z}...
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Confusion in the definition of direct product of finite groups

Let $G$ be a finite group. We will say that $G=A \times B \times C$ if A,B,C are normal in $G$ $A\cap B \cap C ={e}$ $|G|=|A||B||C|$ Is the first condition ok? or should I say $A \times B$ is normal ...
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A condition equivalent to $R$ being a direct summand

If $R\subset S$ are rings, then why is saying that $R$ is a summand of $S$ as an $R$-module the same as saying that there is an $R$-module homomorphism $S\to R$ that fixes all elements of $R$? The ...
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How to determine if $W_1\cap W_2$ is or isn't $0$, and how to then calculate the section

As an example for direct sums in my textbook they have given three vectors contained in the vectorspace $V = \mathbb{R}^3$: $W_1 = \langle(1,0,0)^t,(0,1,0)^t\rangle$ $W_1 = \langle(1,2,3)^t,(2,3,4)^t\...
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104 views

Linear dependence as a binary relation

Elements $a$ and $b$ from an $R$-module $M$ are linearly dependent if there are scalars $x$, $y$ in $R$, $x \neq 0$ or $y \neq 0$, such that $xa = yb$. Let $M$ be a torsion-free module over an ...
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$H$ Hilbert space, $T$ symmetric bounded linear, when is $H=R(T) \oplus N(T)$?

I just saw in an exercise that if I have a prehilbert space $H$ and $T$ a linear, bound and symmetric operator then $R(T)=N(T)^{\perp}$. Now I was asking myself whether $H=R(T) \oplus N(T)$. On wiki I ...
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Sufficiency for proof that if $P \in \mathcal{L}(V)$, such that $P^2 = P$ then $V = \text{null}(P) \oplus \text{range}(P).$

I have seen numerous proofs of this result, and understand why they are true. For instance here and here use the same method - writing $v = Pv + (I - P)v$ and then continuing on in a straightforward ...
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Write a subspace as a Kernel of a linear application

I'd like to discuss the following problem : Write U = { $f \in V | \hspace{0.3 cm}x^{2} | f$},where | means "divides", and $ V = \mathbb{R}_{k}[x]$ As kernel of the linear application : $$ F : \...
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On proposition I.1.2 of “Quantum Groups” by Christian Kassel

I am working through Christian Kassel's textbook on Quantum Groups. The Proposition states that 5 statemens are equivalent. The two I am having trouble with are as follows. 1.For any pair $V'\subset ...
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Application of the cyclic decomposition theorem to a linear transformation

Given a linear transformation $L:V→V$ where $V$ has basis $\{v_1, v_2, v_3\}$ with $L(v_i) = v_i$ for all $i < 3$ and $L(v_3) = v_1 + v_2$, decompose $V$ as a direct sum of $T$-cyclic subspaces as ...
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Show that $R^n/Im(\rho)=R^{n-1}\bigoplus R/2R$, where $R$ is an abelian group and $\rho$ is the following function.

Consider the following group homomorphism $\rho$, where $R$ is an abelian group, \begin{align*} \rho:&R\rightarrow R^n\\ \rho(r)=&(2r,2r,\cdots,2r). \end{align*} Show that $R^n/Im(\rho)=R^{n-...
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Direct sum of non-abelian groups doesn't satisfy the universal property of direct sum.

Let $C$ and $D$ be non-abelian groups. Show that $C\oplus D$ doesn't satisfy the universal property of direct sum. I think I must assume that the universal property is true and then use the free ...
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Extensions of indecomposable modules

Let $R$ be a unital ring. Suppose that $A$, $B$, and $C$ are unitary left $R$-modules such that there exists a non-split exact sequence $$0\to A \overset{\alpha}{\longrightarrow}B\overset{\beta}{\...
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If $AB=0$ for two commuting linear operators and their kernels intersect trivially, do they span the whole vector space?

$\DeclareMathOperator{Ker}{Ker}\DeclareMathOperator{Im}{Im}$ In the finite dimensional case, this is clear by counting dimensions: The image of $B$ must lie completely in the kernel of $A$, and thus ...
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How to show that if $X=M \oplus N$ is a Banach space then $\exists c > 0$ $\forall m\in M \, \forall n\in N: ||m||+||n|| \leq c ||m+n||$? [closed]

Let $X=M \oplus N$, where $X$ is a Banach space and $M$ and $N$ are closed subspaces of $X$. How to prove that $\exists c > 0$ constant such that $\forall m \in M$ and $\forall n \in N$ $$ \left\...
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Finite Length Modules

Let $R$ be a ring. If $M, N$ and $L$ are $R$-modules, with, $\ell(M), \ell(N), \ell(L) < \infty,$ and $M \times N \cong M \times L,$ it is true that $N \cong L?$ The same occurs replacing $\times$...
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Matrices as Outer Direct Sum of Vector Spaces

So studying linear algebra I encountered the outer direct sum and the direct product of collections of vector spaces (which are the same if the collections are finite). When thinking about the uses of ...
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$M_1$, $M_2$ are submodules of a module $M$, then $M = M_1 + M_2$ and $M_1 \cap M_2 = 0$ implies M is isomorphic to $M_1 \oplus M_2$?

I saw the two properties mentioned by the post on Let $R$ be a ring, $M$ an $R$-module, and $A, B ≤ M$ two submodules of $M$ such that $M = A ⊕ B$. Prove that $M/A \cong B$. With intuition I ...
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Does a direct subtraction of vector subspaces make sense?

Consider $E$ a vector space over any field and $U,V$ two subspaces of it, such that $U$ is a subspace of $V$, too. If I want to refer to the subspace $\bar{U}\space\colon\space U \oplus \bar{U}=V$, ...
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If $M$ has a largest proper submodule, then $ M$ is directly indecomposable

How to prove ; "Every module $M$, which has a largest proper submodule or, in the set of non-zero submodules, a smallest submodule, is directly indecomposable?"
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Direct sum of Banach Spaces [closed]

I want to show that direct sum of $c_0$ and $\mathbb K$ is isomorphic to $c_0$. Here $c_0$ is the Banach space of all sequences converge to $0$. Please give me a hint how to define the Isomorphism.
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Clarifying $\oplus$ notation for modules and vector spaces?

I've always been a little confused about this notation. I would really appreciate it if someone could verify that my understanding is correct: The definition of $M_1 \oplus M_2 \cdots \oplus M_k$ ...
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1answer
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A different definition of coproduct in a category

I am following Pavel et al book "Tensor Categories". They write that an additive category is a category $\mathcal{C}$ such that: My problem is with (A3). i know that what they are trying to say is ...
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Let $T: V \to V$ be a linear map such that $T^2-3T+2I=0$.

Let $T: V \to V$ be a linear map such that $T^2-3T+2I=0$, where $I$ is the identity map. question: a) Prove that $V=\ker(T-2I) \oplus\ker(T-I)$ b) let $A$ be an $n \times n$ matrix such that $A^2-...