Questions tagged [direct-sum]
For questions about taking the direct sum of groups and other algebraic structures.
930
questions
0
votes
0
answers
23
views
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 \...
- 1
1
vote
1
answer
26
views
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: ...
- 341
0
votes
1
answer
75
views
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\...
- 91
2
votes
0
answers
37
views
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 ...
2
votes
1
answer
49
views
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 ...
- 23
0
votes
0
answers
32
views
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)...
- 488
0
votes
0
answers
13
views
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 ...
- 262
1
vote
0
answers
31
views
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=...
- 31
3
votes
3
answers
84
views
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 ...
- 571
3
votes
0
answers
26
views
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 ...
- 381
0
votes
0
answers
8
views
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 ...
- 105
0
votes
2
answers
36
views
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 ...
- 269
1
vote
0
answers
30
views
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 ...
- 2,328
2
votes
1
answer
41
views
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 ...
- 53
0
votes
1
answer
43
views
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 ...
0
votes
0
answers
28
views
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=...
- 259
0
votes
0
answers
34
views
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 $\...
- 55
1
vote
2
answers
36
views
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,\ (...
- 71
0
votes
0
answers
18
views
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, ...
- 1,188
0
votes
0
answers
24
views
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 ...
- 2,077
0
votes
1
answer
33
views
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
$$G\cong H\oplus F\quad\text{if and only if}\quad G/H\cong F$$
...
- 493
1
vote
1
answer
37
views
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 ...
- 1,248
0
votes
0
answers
18
views
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_\...
- 5,991
0
votes
1
answer
35
views
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$-...
- 27
2
votes
0
answers
34
views
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 $(...
- 1,570
1
vote
0
answers
43
views
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^...
- 195
0
votes
2
answers
44
views
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$...
- 2,077
1
vote
1
answer
38
views
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 ...
- 525
3
votes
2
answers
83
views
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\...
- 1,072
1
vote
1
answer
52
views
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) \...
- 111
4
votes
1
answer
62
views
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 ...
- 2,100
0
votes
0
answers
38
views
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 \...
- 110
3
votes
1
answer
43
views
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 ...
- 571
-1
votes
1
answer
104
views
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 $...
- 873
1
vote
1
answer
48
views
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 ...
3
votes
1
answer
68
views
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$.
...
- 1,699
3
votes
0
answers
45
views
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 ...
- 1,025
3
votes
2
answers
111
views
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 ...
- 295
1
vote
0
answers
57
views
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 ...
- 165
1
vote
1
answer
82
views
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 ...
- 1,031
0
votes
1
answer
29
views
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)$
...
1
vote
2
answers
46
views
$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\...
- 1,031
1
vote
1
answer
45
views
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$$
Show that this expression is unique, that is, there does not exist other, different linear ...
- 13
0
votes
0
answers
43
views
Direct sum of topological vector space and the convergence of sequence
Suppose $X$ is a topological vector space, $Y$ and $Z$ are two subspaces of $X$, and $X$ is the direct sum of $Y$ and $Z$, write $X=Y\oplus Z$. $\{x_n\}\subset X$ and $x_n=y_n+z_n$ where $y_n\in Y$ ...
- 1
0
votes
0
answers
20
views
When does a graded map of finite dimensional Hilbert space have a graded adjoint?
Suppose I have two finite dimensional graded inner product spaces $$V = \oplus_\alpha V_\alpha \hspace{3em} W = \oplus_\beta W_\beta$$
and a map $f : V \to W$. I can represent $f$ as a direct sum of ...
1
vote
1
answer
42
views
Addition and Tensor product of Vector spaces for beginners : Concrete example
I am looking for a concrete example for expressions like
$$
V_A\otimes V_B = V_C\oplus V_D
$$
that shows explicitly what the basis elements actually look like. My attempt was the following, lets take $...
- 169
1
vote
0
answers
22
views
How can an internal direct sum of 2 abelian groups be a proper subgroup of one of the summands?
I have spent days now trying to understand the proof of an "easy lemma" in Martin Isaacs' book "Algebra: A graduate course". The lemma is 11.10 on page 150, which reads the ...
3
votes
1
answer
91
views
If $A$ is a direct sum of matrix algebra over $C$, what are all finite dimensional simple $A$-modules?
If $A$ is a direct sum of matrix algebra, what are all finite-dim simple $A$-modules?
For simple case of $2$ matrix algebras, consider $A = M_n(\mathbb{C}) \oplus M_m(\mathbb{C})$. Suppose $M$ is a ...
- 33
0
votes
1
answer
36
views
Pairwise disjoint vector spaces whose sum is not direct [closed]
Can we find a collection of vector spaces $U_1, \dots, U_k$ such that $U_i \cap U_j=\{0\}$ for all $i \neq j$, but the sum $U_1+\dots+U_k$ is not a direct sum? I’m not really sure where to start with ...
- 455
0
votes
1
answer
24
views
Transfer from external direct sum to internal direct sum
Let $K$ be a $n$-dimensional vector subspace of $\mathbb R^N$. Then we have the surjective canonical homeomorphism $$f: \mathbb R^N \to \mathbb R^N/K, \quad x \mapsto x+K.$$
We have $\operatorname{ker}...
- 10k