Questions tagged [direct-sum]
For questions about taking the direct sum of groups and other algebraic structures.
0
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65 views
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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1answer
11 views
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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0answers
28 views
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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2answers
39 views
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 ...
2
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2answers
29 views
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 ...
1
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1answer
38 views
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 ...
1
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1answer
25 views
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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0answers
25 views
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 $\mathbb{Z}$-...
1
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1answer
26 views
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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1answer
115 views
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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0answers
26 views
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 ...
2
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2answers
49 views
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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1answer
28 views
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 ...
1
vote
1answer
36 views
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 ...
1
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1answer
61 views
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 ...
0
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1answer
48 views
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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0answers
58 views
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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35 views
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 $...
3
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1answer
131 views
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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0answers
17 views
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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0answers
44 views
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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0answers
38 views
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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2answers
32 views
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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1answer
82 views
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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2answers
46 views
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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votes
1answer
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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0answers
30 views
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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1answer
16 views
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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0answers
28 views
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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1answer
41 views
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\...
1
vote
1answer
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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1answer
48 views
$H$ 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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3answers
63 views
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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1answer
40 views
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 : $$ F : \...
3
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1answer
65 views
On proposition I.1.2 of “Quantum Groups” by Christian Kassel
I am working through Christian Kassel's textbook on Quantum Groups. The Proposition states that 5 statemens are equivalent. The two I am having trouble with are as follows.
1.For any pair $V'\subset ...
2
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0answers
28 views
Application of the cyclic decomposition theorem to a linear transformation
Given a linear transformation $L:V→V$ where $V$ has basis $\{v_1, v_2, v_3\}$ with $L(v_i) = v_i$ for all $i < 3$ and $L(v_3) = v_1 + v_2$, decompose $V$ as a direct sum of $T$-cyclic subspaces as ...
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2answers
34 views
Show that $R^n/Im(\rho)=R^{n-1}\bigoplus R/2R$, where $R$ is an abelian group and $\rho$ is the following function.
Consider the following group homomorphism $\rho$, where $R$ is an abelian group,
\begin{align*}
\rho:&R\rightarrow R^n\\
\rho(r)=&(2r,2r,\cdots,2r).
\end{align*}
Show that $R^n/Im(\rho)=R^{n-...
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1answer
55 views
Direct sum of non-abelian groups doesn't satisfy the universal property of direct sum.
Let $C$ and $D$ be non-abelian groups. Show that $C\oplus D$ doesn't satisfy the universal property of direct sum.
I think I must assume that the universal property is true and then use the free ...
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1answer
40 views
Extensions of indecomposable modules
Let $R$ be a unital ring. Suppose that $A$, $B$, and $C$ are unitary left $R$-modules such that there exists a non-split exact sequence
$$0\to A \overset{\alpha}{\longrightarrow}B\overset{\beta}{\...
3
votes
2answers
78 views
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 ...
2
votes
1answer
34 views
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\...
3
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1answer
29 views
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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0answers
34 views
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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0answers
23 views
$M_1$, $M_2$ 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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1answer
50 views
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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votes
0answers
36 views
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?"
0
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1answer
87 views
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.
0
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0answers
34 views
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 $M_1 \oplus M_2 \cdots \oplus M_k$ ...
2
votes
1answer
46 views
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 ...
5
votes
2answers
232 views
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-...
