Questions tagged [direct-sum]

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

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Limit of components sequences

Let $\mathbb{R}^2 = \mathbb{R}_1 \bigoplus \mathbb{R}_2$ be a direct sum of $\mathbb{R}^2$. Suppose that there are $t_n, t'_n \to +\infty; s_n, s'_n \to s; x_n \to x_1 \in \mathbb{R}_1; x'_n \to x_2 \...
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For a representation $(V, p_V)$ by a finite group and $W = \bigoplus \limits_{i = 1}^{n} V$ calculate $\dim(\text{Hom}_G(V,W))$

Let $(V, p_V)$ be a vector space with a representation by a finite group G. Assume further that $V$ is irreducible and $W = \bigoplus \limits_{i = 1}^{n} V$ with the direct-sum representation. Namely: ...
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How to show that two groups are not isomorphic. [duplicate]

I have learned various theorems that tell me when two groups are isomorphic. For example, if the greatest common divisor of $j$ and $k$ is equal to one, then $\mathbb{Z}_j\oplus\mathbb{Z}_k\cong\...
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Why is it difficult to define a direct integral of Banach spaces or Banach algebras?

In https://en.wikipedia.org/wiki/Direct_integral I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras. Suppose that I want to define a direct integral on either ...
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Need help for understanding a sum of subspaces

I am newbie started learning the linear algebra. It might be dumb question. But I don't understand how the sum of subspace can also be subspace?! So for subset in order to be subspace, It should ...
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A direct sum of Symmetric and Alternating Bilinearforms

Show that the vector space $\text{Bil}(V)$ of all bilinear forms on $V$ can be decomposed in to the direct sum of $\text{Bil}(V)_{\text{sym}} \bigoplus \text{Bil}(V)_{\text{alt}}$, where $\text{Bil}(V)...
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Abelian groups for which every direct sum decomposition contains a Boolean summand

I am interested in the following Question Let $G$ be a countable abelian group with the property that whenever $G\cong H\times K$, at least one of the factor is Boolean. Is it always true that $G\cong ...
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What is the difference between the cartesian product and direct sum of vectors?

My notes give the cartesian product of the sets $X_1, . . . , X_n$ as $$X_1 × · · · × X_n = \{(x_1, . . . , x_n) : x_i ∈ X_i for 1 \le i \le n\}$$ I believe we can think of a vector space $V$ where $V=...
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Are there complements of $\mathbb{Q}$ in $\mathbb{R}$ that are (non-unital) subrings?

Because $(\mathbb{Q}, +)$ is a divisible group, $\mathbb{R} = \mathbb{Q} \times H$, for some subgroup $H$ (if one sees $\mathbb{R}$ as a $\mathbb{Q}$-vector space, for instance one can choose a basis ...
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Splitting Lemma for Vector Bundles

I am asked to solve the following exercise. Let $E = E[M; \pi, \mathbb{R}^n]$,$F = F[M; \pi, \mathbb{R}^m]$, $H = H[M; \pi, \mathbb{R}^k]$ be three smooth vector bundles over $M$ of finite rank $n,m,k ...
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Simple question- Diagonalization and direct sum of image and kernel

Is the following statement somewhat true? A matrix $A\in M_n(\mathbb{F})$ is diagnozable if (or iff) $Ker(A)\bigoplus Im(A)=\mathbb{R}^n$ or $=M_n$ not quite sure on what exactly the result of the ...
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Direct sum of two elements of subspaces of a vector space

Help me to understand what the authors of this paper (p. 3) mean by the direct sum of two elements in a vector space. Let $X$ be a vector space with subspaces $Y$ and $Z$ Definition: X is a direct sum ...
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Proof verification - proving a matrix is diagonalizable using representation theory

Let $A\in M_n(\mathbb{C})$ be a matrix s.t $A^N=I_n$. Prove, using representation theory, that $A$ is diagonalizable. My attempt: We look at $G=\langle A\rangle\subset GL_n(\mathbb{C})$. This is a ...
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Direct sum of reproducing kernel Hilbert spaces (RKHS)

I am currently diving into the theory of reproducing kernel Hilbert spaces and am just at the beginning of understanding the background of reproducing kernels. I have stumbled upon the following ...
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Making sense of a derivative on direct sum

Consider $\mathbb{R}^n=E(t)\bigoplus F(t) $ be a continuous splitting. Is that possible to define a derivative on the direct sum such that $$ (x(t)\bigoplus y(t))'=x'(t)\bigoplus y'(t) $$ The reason ...
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Showing that $\mathbb{F}^n = \mathrm{span}\{\sum_{i=1}^ne_i\}\oplus \mathrm{span}\{\sum_{i=1}^nc_ie_i\mid \sum_{i=1}^nc_i = 0\}$

Let $\mathbb{F}^n$ be a vector space over the field $\mathbb{F}$ with a basis $\{e_1,\dots,e_n\}$. I'm trying to show that $\mathbb{F}^n = \mathrm{span}\{\sum_{i=1}^ne_i\}\oplus \mathrm{span}\{\sum_{i=...
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Prove that $V=U_1\oplus\cdots\oplus U_k$ iff $v=u_1+\cdots+u_k$

Let $U_1,U_2,...,U_k$ be subspaces of vector space $V$. Prove that $V=U_{1}\oplus U_{2}\oplus...\oplus U_{k}$ iff every vector $v\in V$ can be written uniquely as $v=u_{1}+u_{2}+...+u_{k}$ while $\...
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To show: Direct sum of subspaces $M$ and $N$ in $R^2$ equals $R^2$, where $M=\{(x,y)\in R^2|2x+y=0 \}$ and $N=\{(x,y)\in R^2|x-y=0 \}$

If $M=\{(x,y)\in \mathbb{R}^2|2x+y=0 \}$ and $N=\{(x,y)\in \mathbb{R}^2|x-y=0 \}$. Show that $M+N=\mathbb{R}^2$ My Attempt: $M+N=\{(x,y)\in \mathbb{R}^2|(x,y)=(x_1,y_1)+(x_2,y_2);\ (x_1,y_1)\in M,\ (...
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Is there any factorization of Leibniz algebras?

A semigroup S is factorisable if there are subsemigroups A and B such that S = AB. In the case of Leibniz algebras, can we say that a Leibniz algebra is a direct sum of two subalgebras? Any reference, ...
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Is $V = \text{null}(A - \lambda I) \oplus \text{range}(A - \lambda I)$ for eigenvalue $\lambda$?

For $A: V \rightarrow V$ a linear operator on $V$ a finite dimensional vector space, it is true that $\text{null}(A - \lambda I)$ and $\text{range}(A - \lambda I)$ are $A$-invariant, right? It is also ...
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Direct product and quotient groups

I am trying to convince myself that the following statement is true: Let $G, F$ be two group and $H$ be a normal subgroup of $G$. Then $$G\cong H\oplus F\quad\text{if and only if}\quad G/H\cong F$$ ...
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What is the relationship between projective subspaces of $\mathbb{P}(V \oplus \mathbb{F})$ and affine subspaces of $V$? What is $V \oplus \mathbb{F}$?

Define an embedding $V \subset \mathbb{P}(V \oplus \mathbb{F})$. What is the relationship between projective subspaces of $\mathbb{P}(V \oplus \mathbb{F})$ and affine subspaces of $V$? This question ...
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Tensor - sum distributivity with common index

I am well aware of the distributivity of tensor product over direct sum for $R$-modules over a commutative ring. I see how it works for direct sums and tensor products of the form $$(\oplus_\alpha M_\...
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A problem about cyclic subspaces and minimal polynomial

Let $\alpha$ be a linear operator on a vector space $V$, and supoose that $V$ is $\alpha$-cyclic, say generated by $v\in V$. Suppose further that $V=U_1\bigoplus U_2$ for non-trivial $\alpha$-...
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Direct sum of direct integrals

Let $(X,\mu)$ be a measure space, and $(H_x)_{x \in X}$ is a "field" of Hilbert spaces, one can form the direct integral $\int^{\oplus} H_x d\mu(x)$ which is a Hilbert space as well. When $(...
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Cohomology ring of disjoint union

It's a well-known fact that $H^*(\bigsqcup_{\alpha}X_{\alpha};R)\xrightarrow{\cong}\prod_{\alpha}H^*(X_{\alpha};R)$ that is also in Hatcher Chapter 3, example 3.13. My attempt by definition is that $H^...
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In linear algebra, does $A \perp B$ mean the same thing as $A = B^\perp$?

I'm having a hard time wrapping my head around orthogonal complements. I think my brain just rejects the notation, for whatever reason. If I could write $A \perp B$ and read it as, "The subset $A$...
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1 answer
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Two competing definitions of the direct sum of vector spaces

There seem to be two competing definitions of the direct sum of vector spaces. The first one characterises it as the same as the Cartesian product for a finite number of vector spaces, and for an ...
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2 answers
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What is the kernel of this map $\Phi: F_2 \to \mathbb{Z}_2 \oplus \mathbb{Z}_3$?

What is the kernel $K\leq F_2 = \langle a,b \rangle$ of this map $\Phi: F_2 \to \mathbb{Z}_2 \oplus \mathbb{Z}_3$ given by $a \mapsto (1+2\mathbb{Z},0+3\mathbb{Z})$ and $b\mapsto (0+2\mathbb{Z}, 1+3\...
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1 answer
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Direct sum of images of endomorphisms

Let $f, g: V\to V$ be $K$-linear functions, $V$ a $K$-vector space for some field $K$. Show, that if $V=\text{im}(f)+\text{im}(g)=\ker(f)+\ker(g)$ and $V$ has finite dimension, then $V=\text{im}(f) \...
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4 votes
1 answer
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Direct Sum of Groups - are there 2 types of direct sums?

This has been a point of confusion for quite some time now for me. I have come across two different definitions of the direct sum on groups, each with a different notation for the symbol. These two ...
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Tensor distributes over direct sum

Let $A$ be a commutative ring with identity, and let $(M_i)_{i \in I}$ be a family of $A$-modules (for some index set $I$). I am trying to show that for all $A$-modules $N$, $N \otimes (\bigoplus_{i \...
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Does the torsion subgroup of a torus have more than one complement?

Let $A$ be a torus (compact connected abelian Lie group) and $T$ its torsion subgroup. Since $T$ is divisible, then $A = T \times H$ for some torsion-free subgroup $H$. Is $H$ unique? (I know in ...
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If $N \leq M$, there exist free modules $F_N \leq N$, $F_M \leq M$ with $N = F_N \oplus N_{tor}$, $M = F_M \oplus M_{tor}$ with $F_N \leq F_M$

Let $R$ be a principal ideal domain and $M$ a finitely generated $R$-module. Furthermore, let $N$ be a submodule of $M$. Prove or disprove: there exist free submodules $F_N \leq N$, $F_M \leq M$ with $...
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1 answer
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Prove a projection transformation is linear

Let $F$ and $G$ be subspaces of a vector space $V$ of finite dimension, such that they satisfy $F \oplus G = V$. Let $P:V \rightarrow V$ be a function that satisfies: i) $P(v) \in F$ ii) $v - P(v) \in ...
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3 votes
1 answer
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Infinite direct sum not isomorphic to cartesian product?

Let $\{V_i: i \in I\}$ be an infinite collection of non-trivial (i.e. not dimension $0$) vector spaces. I am trying to prove that $\bigoplus_{i \in I} V_i$ is not isomorphic to $\Pi_{i \in I} V_i$. ...
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For $P$ the set of all primes, ${\rm Hom}(\Bbb Z,\sum_{p\in P}\Bbb Z_p)\not\cong\prod_{p\in P}{\rm Hom}(\Bbb Z,\Bbb Z_p)$

${\rm Hom}(\mathbb Z, \sum_{p\in P}\mathbb Z_p)$ and $\prod_{p\in P}{\rm Hom}(\mathbb Z, \mathbb Z_p)$ are not isomorphic where $P$ is the set of all primes. I was checking the elements of each to see ...
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2 answers
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Why is the space of differential forms $\bigoplus_{p=0}^n \Lambda_x^p$?

In Wald's book "General Relativity", the space $\Lambda_x$ of differential forms at a point $x$ is worked out in the following manner: Let $M$ be an $n$-dimensional manifold. The vector ...
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Isomorphism between the direct sum of submodules and generated submodule

I am reading Dummit & Foote Chapter 10 and stuck on this exercise ($R$ is a ring with $1$ and $M$ is a left $R$-module): I am unsure about how to proceed in (i) => (ii). I assume the direct ...
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Prove that additive functor preserves products and coproducts

Let $\cal A,B$ be additive categories and $F:\cal A\rightarrow B$ be an additive functor. Show that $F$ preserves products and coproducts. Since product and coproduct of a pair $A,B$ of objects in an ...
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Index notation in question about distribuitivity of tensor product over direct sum

Thanks to @peek-a-boo I edited my question In this post Tensor product and direct sum the author has the map $\varphi : (\bigoplus_\alpha M_\alpha)\times N\to \bigoplus_\alpha (M_\alpha\otimes N)$ ...
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2 answers
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$p_Ai_B=0$ and $p_Bi_A=0$ in additive category

Let $\cal A$ be an additive category. Then for any $A,B\in\textrm{Ob}(\cal A)$ the direct sum $A\oplus B$ is both their product and their coproduct. Let $i_A:A\rightarrow A\oplus B$ and $i_B:B\...
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1 vote
1 answer
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is linear combination unique for each vector?

each vector $\vec x$∈$R^n$ can be expressed as a sum $$\vec x = \vec x_1\vec e_1+···+\vec x_n\vec e_n$$ Show that this expression is unique, that is, there does not exist other, different linear ...
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Direct sum of topological vector space and the convergence of sequence

Suppose $X$ is a topological vector space, $Y$ and $Z$ are two subspaces of $X$, and $X$ is the direct sum of $Y$ and $Z$, write $X=Y\oplus Z$. $\{x_n\}\subset X$ and $x_n=y_n+z_n$ where $y_n\in Y$ ...
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When does a graded map of finite dimensional Hilbert space have a graded adjoint?

Suppose I have two finite dimensional graded inner product spaces $$V = \oplus_\alpha V_\alpha \hspace{3em} W = \oplus_\beta W_\beta$$ and a map $f : V \to W$. I can represent $f$ as a direct sum of ...
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1 vote
1 answer
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Addition and Tensor product of Vector spaces for beginners : Concrete example

I am looking for a concrete example for expressions like $$ V_A\otimes V_B = V_C\oplus V_D $$ that shows explicitly what the basis elements actually look like. My attempt was the following, lets take $...
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How can an internal direct sum of 2 abelian groups be a proper subgroup of one of the summands?

I have spent days now trying to understand the proof of an "easy lemma" in Martin Isaacs' book "Algebra: A graduate course". The lemma is 11.10 on page 150, which reads the ...
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1 answer
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If $A$ is a direct sum of matrix algebra over $C$, what are all finite dimensional simple $A$-modules?

If $A$ is a direct sum of matrix algebra, what are all finite-dim simple $A$-modules? For simple case of $2$ matrix algebras, consider $A = M_n(\mathbb{C}) \oplus M_m(\mathbb{C})$. Suppose $M$ is a ...
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1 answer
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Pairwise disjoint vector spaces whose sum is not direct [closed]

Can we find a collection of vector spaces $U_1, \dots, U_k$ such that $U_i \cap U_j=\{0\}$ for all $i \neq j$, but the sum $U_1+\dots+U_k$ is not a direct sum? I’m not really sure where to start with ...
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Transfer from external direct sum to internal direct sum

Let $K$ be a $n$-dimensional vector subspace of $\mathbb R^N$. Then we have the surjective canonical homeomorphism $$f: \mathbb R^N \to \mathbb R^N/K, \quad x \mapsto x+K.$$ We have $\operatorname{ker}...
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