Questions tagged [direct-sum]
For questions about taking the direct sum of groups and other algebraic structures.
1,008
questions
3
votes
1
answer
105
views
Does the direct sum have a universal property in the category of groups?
In other categories, like modules, the direct sum is the coproduct, giving it a neat category theory description. The direct product of groups, which is very similar to it, is the product in the ...
- 2,644
1
vote
0
answers
23
views
Infinite decomposition of a Hilbert space
Let $H$ be a Hilbert space. Suppose we have $H_{n}\oplus G_{n}=H$ for all $%
n\geq 0$ so that $\left( H_{n}\right) _{n\geq 0}$ is an increasing sequence
of closed subspaces and $\left( G_{n}\right) _{...
- 1,001
0
votes
0
answers
33
views
What is the precise meaning of $c_\infty \cong c_0 \oplus \mathbb{C}$?
I am supposed to show that $c_\infty \cong c_0 \oplus \mathbb{C}$, where
$$
c_\infty := \{(a_n)_{n \in \mathbb{N}}| a_n \in \mathbb{C}, a_n \to a \in \mathbb{C}\}, \\
c_0 := \{(a_n)_{n \in \mathbb{N}}|...
- 383
1
vote
1
answer
31
views
Hyperplanes question in Tu's book on Manifolds
I am trying an exercise from Tu's book "An introduction to Manifolds". Specifically, Subchapter 1.3, exercise 2 (b).
I would like, before continuing with my question, to discuss something ...
- 189
1
vote
0
answers
61
views
When does $A^\perp = 0$ imply that $A=X$ for a subspace $A\subset X$ of a Banach space
Setup
Consider the Banach space $X=\mathcal C^1([0,1],\mathbb R^n)$ with norm
$$
\|f\|=\sup |f|+\sup |\dot f|.
$$
This banach space has a symmetric bilinear positive definite form
$$
b(f,g)=\int_0^1\...
- 611
3
votes
1
answer
61
views
If $A$ and $A^\perp$ are closed then $A\oplus A^\perp=B$ for a Banach space $B$
Let $b$ be a bounded, positive definite and symmetric bilinear form on a Banach space $B$.
I want to prove the following:
Let $A\subset B$ be a closed subspace such that $A^\perp$ is closed too. Then $...
- 611
0
votes
1
answer
84
views
Decomposition of TTM into HM and VM
Why is it that if I have a smooth manifold and a connection map $K$, defined below, is it the case that it induces a decomposition of the tangent space to the tangent space to the manifold, given as ...
- 938
1
vote
1
answer
41
views
Are there internal direct products of vector spaces?
When I learned linear algebra I was a bit confused about the notation $A \oplus B$, the direct sum of two vector spaces $A, B$ over some field $K$. As a set, $A \oplus B$ is nothing else than the ...
- 3,355
2
votes
1
answer
49
views
$N_i$'s independent $\iff$ $\oplus_i N_i\cong\sum_i N_i$?
Notation and conventions: I will use $\oplus$ for the "external direct sums" only. All the modules will be over a commutative ring with unity, $A$.
Problem: In the proof of one of the ...
- 2,638
1
vote
1
answer
86
views
Theorem 5, Section 7.2 of Hoffman’s Linear Algebra
Definition: $A\in M_{n\times n}(F)$ is in rational form if $$A=\begin{bmatrix} A_1& & \\ & \ddots & \\ & & A_r\\ \end{bmatrix}$$ where $A_i$ is companion matrix of non scalar ...
- 3,180
1
vote
0
answers
22
views
Direct Sum of Topological $R$-modules
Please help me. I am proving the following property: Let $R$ be a topological ring and $\{A_{\lambda}\}_{\lambda\in\Lambda}$ be a family of topological $R$-modules. Let $B_{\lambda}$ be an ...
- 51
2
votes
1
answer
41
views
Is a group complement of a subspace of a vector space necessarily a subspace?
Let $W$ be a vector space. Let $U$ be a subspace and $V$ be a subgroup of $W$, and assume that $W$ is the direct sum (as groups!) of $U$ and $V$. Does it follow that $V$ is a subspace, such that in ...
- 6,388
0
votes
1
answer
34
views
Endomorphisms between direct product/sum of algebraic objects?
My motivation is this Wikipedia article.
Suppose $R$ be a ring with unity and $M,N$ be $R$-modules.
Take their direct sum/product $P=M \oplus N$. So $P$ is also a $R$-module. Consider the respective ...
- 165
1
vote
1
answer
39
views
About the definition of direct sum of Hilbert spaces
Let $\{\mathscr{H}_{i}\}_{i\in I}$ be a family of Hilbert spaces. Their direct sum, denoted by:
$$\mathscr{H} = \bigoplus_{i} \mathscr{H}_{i},$$
is defined as the space of all functions $\psi: I \to \...
- 1,601
3
votes
0
answers
55
views
Exercise 11, Section 6.6 of Hoffman’s Linear Algebra
Let $V$ be a vector space, let $W_1, \ldots, W_k$ be subspaces of $V$, and let
$$V_j = W_1 + \cdots + W_{j-1} + W_{j+1} + \cdots + W_k.$$ Suppose that $V = W_1 \oplus \cdots \oplus W_k$. Prove that ...
- 3,180
0
votes
1
answer
42
views
What are $e_1, e_2, h_1$ and $h_2 \ ?$
Let $W$ be a finite dimensional complex vector space and $V$ and $U$ be subspaces of $W$ with $V = \mathbb C e \oplus \mathbb C h$ and $U = \mathbb C f \oplus \mathbb C h,$ for some $e,f,h \in W.$ ...
- 2,712
0
votes
0
answers
35
views
Remarks of Theorem 11, Section 6.7 of Hoffman’s Linear Algebra
Following are remarks of theorem 11 section 6.7 :
(i) One of the pleasant features of the decomposition $T=c_1E_1+…+c_kE_k$ is that if $g$ is any polynomial over the field $F$, then $g(T)=g(c_1)E_1+…+...
- 3,180
5
votes
1
answer
64
views
The functor $\mathrm{Hom}(A,-)$ cannot commute with arbitrary direct sums for infinitely generated projective module $A$
It is easy to see the functor $\mathrm{Hom}(A,-)$ commutes with every arbitrary direct sum (i.e. $\mathrm{Hom} (A,\oplus_{i\in I} N_i)=\oplus_{i\in I}\mathrm{Hom}(A,N_i)$) for finitely generated ...
- 341
0
votes
0
answers
39
views
Theorem 11, Section 6.7 of Hoffman’s Linear Algebra
Let $T$ be a linear operator on a finite-dimensional space $V$.
If $T$ is diagonalizable and if $c_1,…, c_k$ are the distinct characteristic values of $T$, then there exist linear operators $E_1,…,E_k$...
- 3,180
0
votes
0
answers
37
views
Exercise 4, Section 6.7 of Hoffman’s Linear Algebra
Let $T$ be a linear operator on $V$. Suppose $V=W_1\oplus … \oplus W_k$,where each $W_i$ is invariant under $T$. Let $T_i$ be the induced (restriction) operator on $W_i$.
(a) Prove that $\det (T)=\det ...
- 3,180
0
votes
1
answer
18
views
Determine a percentage of an unfilled quota
I am having a bit of a computer programming challenge, c#, however the actual issue I believe stems from myself not understanding the maths behing it, hence me posting here...
So my challenge.
We have ...
- 103
0
votes
0
answers
60
views
Theorem 10, Section 6.7 of Hoffman’s Linear Algebra
Theorem 9: $V=W_1\oplus …\oplus W_k$$\iff$$\exists E_1,…,E_k\in L(V,V)$ such that
(i) each $E_i$ is projection ($E_i^2=E_i$)
(ii) $E_iE_j=0$, if $i\neq j$
(iii) $I=E_1+…+E_k$
(iv) $R_{E_i}=W_i$.
...
- 3,180
0
votes
1
answer
20
views
range of continous linear operator between banach spaces with closed algebraic complement
Let X, Y be Banach spaces, and let A ∈ L(X, Y ).
Suppose that there exists a closed subspace W ⊂ Y so that Y is the algebraic direct
sum of ran A = A(X) and W. (That is, every y ∈ Y can be written as ...
- 1
0
votes
1
answer
61
views
Exercise 2, Section 6.6 of Hoffman’s Linear Algebra
Let $V$ be a finite-dimensional vector space and let $W_1,…,W_k$ be subspaces of $V$ such that $V=W_1+…+W_k$ and $\dim(V)=\dim(W_1)+…+\dim (W_k)$. Prove that $V=W_1\oplus …\oplus W_k$.
Approach (1): ...
- 3,180
0
votes
0
answers
30
views
A complex involving an infinite direct sum
Let $A,B_i,C$, for $i \in I$ an infinite index set, be abelian groups such that we have group homomorphisms
$$r_i: A \rightarrow B_i, \,\,\, s_i: B_i \rightarrow C$$
and a complex
$$A \xrightarrow{\...
- 1,052
1
vote
2
answers
33
views
Decomposing the topology of a normed linear space as the product topology of its subspaces
Let's take a normed linear space $V$ and its subspaces $U$ and $W$ such that $V = U\oplus W$. Now, $U$, $W$ are normed linear spaces in their own right. It's easily shown that $U\oplus W\cong U\times ...
- 2,638
2
votes
2
answers
116
views
Linear transformation and direct sum
When given $T:V \to V$ such that $T^2=I$, prove that $U,W \subseteq V$ subspaces exist such that:
$$V=U\oplus W$$
$$T(u)=u, \forall u \in U$$
$$T(w)=-w, \forall w \in W$$
My first thought was to ...
- 43
3
votes
2
answers
82
views
Prove that if $\dim V= n$ and $S$ is a subspace (of dimension $n-1$), then if $v$ doesn't belong to $S$: $\langle v \rangle + S$ is a direct sum
I was doing direct sum exercises and I found this problem:
Assume you have a $k$-vector space $V$ of dimension $n$ and $S$ subspace (of $V$) of dimension $n-1$.
I) Prove that if $v$ doesn't belong to ...
- 317
0
votes
1
answer
69
views
Extending a basis to make a direct sum
Let $V$ be a finite-dimensional vector space. Let $U$ be a subspace of $V$. If I extend a basis $\{v_1, ... , v_m\}$ for $U$ to a basis $\{v_1, ... , v_n\}$ for $V$, does this then imply that $V=U\...
- 303
1
vote
1
answer
165
views
How do I prove that if $A$ and $B$ are complementary then $A^\perp$ and $B^\perp$ are also complementary in $\mathcal{H}$ Hilbert space?
I have the following problem: I know that $A$ and $B$ are two closed complementary subspaces in an Hilbert space $\mathcal{H}$, i.e. $\mathcal{H}=A\oplus B$ (where $\oplus$ is the direct sum of two ...
- 11
1
vote
1
answer
37
views
Kronecker product between a matrix and a direct sum
Is there any way to simplify this matrix expression, using something like a distributive property?
$A \otimes \left( M_1 \oplus M_2 \right)$
where $\otimes$ is the Kronecker product and $\oplus$ is ...
- 352
4
votes
1
answer
137
views
Proving existence of matrix N such that M = MNM
Doing practice problems for my qualifying exam and am a bit stumped by the following:
Let $n \ge 1$, $F$ be a field, and let $M$ be a matrix in $M_n(F)$. Show that there exists an $N \in M_n(F)$ such ...
- 129
1
vote
1
answer
42
views
Find a subspace $W$ of $F^4$ such that $F^4=U \oplus W$
I have to find a subspace $W$ of $F^4$ such that $F^4=U \oplus W$.
$U:=\{(x,x,y,y) \in F^4:x,y \in F\}$
I found the topic with the same problem. The solution was:
$W:=\{(0,b-a,c-d,0)\in F^4:a,b,c,d \...
- 618
3
votes
0
answers
38
views
For $K\oplus L$ if $N$ is normal in $L$ then $(K\oplus L) / N = K\oplus (L/N)$
I'm a bit new to direct sums etc., so for the purpose of convincing myself, formally, that $(\mathbb{Z} \oplus \mathbb{Z})/\langle (2,2)\rangle = \mathbb{Z} \oplus \mathbb{Z}_2$, using the basis $\{(1,...
- 1,555
1
vote
1
answer
77
views
Difference between direct sum of vector spaces and direct sum of representations.
Let $G$ be a group. Let $V$ and $W$ be vector spaces which have the structures of $G$-modules. If $U \simeq V \oplus W$ as vector spaces then can't we say that they are same as $G$-modules? My teacher ...
- 2,712
0
votes
0
answers
8
views
Determine those s ∈ $\mathbb R$ for which $U_1$ + $T_s^+$ matches with $\mathbb R^\mathbb R$
For any s ∈ $\mathbb R$ the subsets of $\mathbb R^\mathbb R$ are difined:
$T_s^+$ := {f ∈ $\mathbb R^\mathbb R$ | f(x) = 0 for all x $\ge$ s},
$T_s^-$ := {f ∈ $\mathbb R^\mathbb R$ | f(x) = 0 for all ...
- 39
0
votes
0
answers
61
views
Determine bases of U1 + U2, U2 + U3, U1 + U2 + U3. Which of these sums are direct
Three subspaces of R^4×1; U1 =[a], U2 =[{b,c}], U3 =[{b, a}]
a = (0, 1, 1, 0)^T , b = (1, 0, 0, 1)^T , c = (0, 1, 0, 0)^T.
Determine bases of U1 + U2, U2 + U3, U1 + U2 + U3. Which of these sums are ...
- 39
0
votes
1
answer
40
views
Can linearity on subspaces imply linearity on their direct sum?
I read an interesting fact while learning calculus of functions of several variables: for fixed bases in $\mathbb{R}^m$ and $\mathbb{R}^n$, the linear mapping $L:\mathbb{R}^m \rightarrow \mathbb{R}^n$ ...
- 57
2
votes
1
answer
55
views
Cross-multiplication theorem of quotient modules.
We know that if $M$ is an $R$-module and $N\subset M$ is a submodule then $M/N\simeq M'$ may not imply $M\simeq N\oplus M'$.In our module theory class,it has been proved that if $M/N\simeq R^k$ then $...
- 2,957
0
votes
0
answers
51
views
Direct sum of two Hilbert spaces is a Hilbert space
I want to prove that the direct sum of two (complex) Hilbert spaces is a Hilbert space.
I've shown that we have an inner product and also shown norm however I have trouble to show converges. We define ...
- 490
2
votes
2
answers
68
views
I need to prove that $Ker(u - Id_E)\oplus Ker(u - aId_E)\oplus Ker(u - a^2Id_E)=E$
Let be E a $\mathbb{C}$ vectorial space and $u\in\mathcal{L}(E,E)$ a linear operator such that $u^3 = Id_E$. Prove that
$$E = Ker(u - Id_E)\oplus Ker(u - aId_E)\oplus Ker(u - a^2Id_E)$$
where $a = -\...
- 57
1
vote
1
answer
73
views
Spectrum in terms of two spectra in bounded Hilbert spaces.
If we have $H_1,H_2$ as hilbert spaces, then
$$H_1\oplus H_2:=\{(h,k):h\in H_1, k\in H_2\}$$
then it is also a Hilbert space with the following,
$$(h_1,k_1)+(h_2,k_2):=(h_1+h_2,k_1+k_2)$$ and $$\...
- 534
0
votes
0
answers
44
views
Example to definition
A direct sum decomposition is $S=\bigoplus_{a \ge 0} S_a$ (d is degree of forms homegeneous polynomial). I try to find and example of this definition, because I need to understand it.
Can I take ...
- 25
1
vote
2
answers
90
views
Spectrum of direct sum of bounded operators
I recently learned something about spectrum in functional analysis and saw some examples. However I struggling with this when trying to understand how it can be used for the following example ...
- 534
0
votes
1
answer
33
views
Using the First Isomorphism Theorem to find a linear transformation
For a vector space V with subspaces U and W, how can we find a suitable linear transformation that by the first isomorphism theorem produces (U + W)/W $\simeq$ U/(U $\displaystyle \cap$ W)?
I gather ...
- 49
0
votes
1
answer
79
views
Confusion about the direct sum of abelian/nonabelian groups [closed]
I know that there have been many questions on this site about the relationship between the direct product and direct sum of groups. But it seems they don't address the specific issue that I want to ...
- 11
4
votes
1
answer
139
views
Norm of bounded operators of direct sum
Yesterday I posted this one regarding direct sum on Hilbert spaces $H_1$ and $H_2$ - have a look!
Direct sum of two Hilbert spaces is a inner product.
I am studying bounded operators and I just want ...
- 534
4
votes
0
answers
83
views
Direct sum of two Hilbert spaces is a inner product.
Today I am working with Functional Analysis regarding Hilbert Spaces from the notes. Let $H_1$ and $H_2$ be Hilbert Spaces. The direct sum $H_1\oplus H_2$ is the vector spaces $H_1\times H_2$ with the ...
- 534
0
votes
1
answer
53
views
Show that $\mathbb{R}^n$ is a direct sum of lines and planes
Let $A \in M_n(\mathbb{R})$ be a matrix which is diagonalizable in
$M_n(\mathbb{C})$. We note $u_{A, \mathbb{R}}: \mathbb{R}^n
\rightarrow \mathbb{R}^n$ the endomorphism defined by the left
...
- 217
0
votes
0
answers
28
views
A decomposition of a vector space using a nilpotent linear map
Let $E$ be a vector space over a field $F$ and $u$ a linear map $E \rightarrow E$ which is nilpotent, ie, $u^p = 0$. I want to show that there exists a decomposition $E = F_1 \oplus ... \oplus F_p$.
...
- 348