Questions tagged [direct-sum]
For questions about taking the direct sum of groups and other algebraic structures.
0
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1answer
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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 ...
0
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1answer
30 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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0answers
14 views
Exercise about a family of homomorphisms between abelian groups
It is considered the category $\mathscr{Ab}$ of abelian groups; a family of homomorphisms: $\{ f_i : M_i \rightarrow N_i | i \in I, M_i,N_i \in \mathscr{Ab} \}$ and constructions $\prod_i f_i$, $\...
1
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1answer
56 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
41 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 $\mathbb{Z}$ ...
2
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0answers
48 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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0answers
27 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
67 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 ...
0
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0answers
6 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.
(...
1
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0answers
32 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
32 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 ...
0
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1answer
79 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 $...
0
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2answers
42 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 ...
0
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1answer
33 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 ...
0
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0answers
27 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}...
0
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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 ...
0
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0answers
25 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
100 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 ...
3
votes
1answer
44 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 ...
2
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3answers
51 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 ...
1
vote
1answer
39 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 : $$ F : \...
3
votes
1answer
63 views
On proposition I.1.2 of “Quantum Groups” by Christian Kassel
I am working through Christian Kassel's textbook on Quantum Groups. The Proposition states that 5 statemens are equivalent. The two I am having trouble with are as follows.
1.For any pair $V'\subset ...
1
vote
0answers
15 views
Application of the cyclic decomposition theorem to a linear transformation
Given a linear transformation $L:V→V$ where $V$ has basis $\{v_1, v_2, v_3\}$ with $L(v_i) = v_i$ for all $i < 3$ and $L(v_3) = v_1 + v_2$, decompose $V$ as a direct sum of $T$-cyclic subspaces as ...
2
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2answers
33 views
Show that $R^n/Im(\rho)=R^{n-1}\bigoplus R/2R$, where $R$ is an abelian group and $\rho$ is the following function.
Consider the following group homomorphism $\rho$, where $R$ is an abelian group,
\begin{align*}
\rho:&R\rightarrow R^n\\
\rho(r)=&(2r,2r,\cdots,2r).
\end{align*}
Show that $R^n/Im(\rho)=R^{n-...
0
votes
1answer
50 views
Direct sum of non-abelian groups doesn't satisfy the universal property of direct sum.
Let $C$ and $D$ be non-abelian groups. Show that $C\oplus D$ doesn't satisfy the universal property of direct sum.
I think I must assume that the universal property is true and then use the free ...
0
votes
1answer
35 views
Extensions of indecomposable modules
Let $R$ be a unital ring. Suppose that $A$, $B$, and $C$ are unitary left $R$-modules such that there exists a non-split exact sequence
$$0\to A \overset{\alpha}{\longrightarrow}B\overset{\beta}{\...
3
votes
2answers
75 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
31 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
votes
1answer
28 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
27 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 ...
0
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0answers
20 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
40 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$, ...
0
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0answers
35 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
50 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.
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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
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1answer
44 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
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2answers
194 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-...
1
vote
1answer
36 views
If $S_1,S_2,S_3$ have direct sum, then, $S_1 = (S_1+S_2) \cap (S_1+S_3)$?
If $S_1,S_2,S_3$ have direct sum, then, $S_1 = (S_1+S_2) \cap (S_1+S_3)$?
I tried this way:
Call $\mathcal F_1$ a family of vectors that generates $S_1$.
Call $\mathcal F_2$ a family of vectors ...
0
votes
1answer
40 views
Direct Sum and Diagonalization with a Linear Map
Let $B$$:$ $W$ $→$ $W$ be a linear map such that $B^2-3B+2I = 0$, where $I$ is the identity map on $W$.
a) Prove that $W$ $=$ $Ker(B - 2I)$ $⊕$ $Ker(B - I)$.
b) Let $M$ be an $n$ x $n$ matrix such ...
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votes
1answer
52 views
Let $R$ be a ring, $M$ an $R$-module, and $A, B ≤ M$ two submodules of $M$ such that $M = A ⊕ B$. Prove that $M/A \cong B$. [closed]
Let $R$ be a ring, $M$ an $R$-module, and $A, B ≤ M$ two submodules of $M$ such that $M = A ⊕ B$. Prove that $M/A \cong B$.
1
vote
1answer
72 views
Associativity of direct sums
Given three vector spaces U, V, and W, which aren't necessarily subspaces of a common vector space, I have to prove that (U $\oplus V) \oplus W \cong U \oplus (V \oplus$ W). I don't even know how I ...
0
votes
2answers
18 views
Question About Direct Sums and Dimension
this was a hw question given in class today, but I am not sure where to begin the proof. There are so many theorems that we went over today, I'm not sure which ones are applicable, and which ones to ...
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5answers
46 views
How to find the sum of $n(n+1)$, $(2n-1)$ and $(3n-2)$
How can I find the next sums?
$$\sum_{k=0}^n k(k+1)$$
$$\sum_{k=0}^n (2k-1)$$
$$\sum_{k=0}^n (3k-2)$$
How can I find their general formula? Maybe don't just lay it all out for me, but tell me how ...
0
votes
0answers
26 views
When are pure subgroups direct summands?
I am working with pure subgroups and I am interested in their relations with direct summands. The Wikipedia article on pure subgroups states that "Under certain mild conditions, pure subgroups are ...
0
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1answer
22 views
How do I interpret *all factors have a common finite exponent* in this context?
"The product of infinitely many torsion groups will no longer be a torsion group unless all factors have a common finite exponent (which is not the case if we take Prüfer groups)."
How do I interpret ...
1
vote
1answer
56 views
If $U =\{ f \in P_3| f(-1)=f(1)=0\}$ Then is $P_3 = U⊕P_2$?
I know that I want to be able to show that $U\cap P_2= \{0\}$ I was able to work out that given the conditions, an element of U should be of the form $ax^3 + bx^2 -ax -b$. Is this correct? If so, how ...
0
votes
0answers
26 views
Proving/disproving a direct sum statement
Let $U ={f \epsilon P3|f(-1)=f(1)=0}$. Prove/disprove:
a) P3= U ⊕ P2
b) P3= U ⊕ P1
My work:
For the first part we know that that -1 and 1 are solutions, so that would give the general form of an ...
2
votes
1answer
26 views
Proof that $fV=fV_1\bigoplus \cdots \bigoplus fV_k$
I'm asked to prove that if $T$ is a linear operator on the vector space $V$, with $V=\bigoplus_{i=1}^k V_i$ ($V_i$ being $T$-invariant) and $f$ is a polynomial over $F$, and we define $f \alpha=f(T)\...
