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Questions tagged [direct-sum]

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

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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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38 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 $V = \mathbb{R}^3$: $W_1 = \langle(1,0,0)^t,(0,1,0)^t\rangle$ $W_1 = \langle(1,2,3)^t,(2,3,4)^t\...
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$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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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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35 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 : \...
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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 ...
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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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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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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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30 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}{\...
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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 ...
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1answer
30 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\...
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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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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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$M_1$, $M_2$ 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
31 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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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?"
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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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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 $M_1 \oplus M_2 \cdots \oplus M_k$ ...
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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 ...
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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-...
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1answer
35 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 ...
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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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45 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 $M/A \cong B$.
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1answer
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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1answer
55 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 ...
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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 ...
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1answer
25 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)\...
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43 views

summation properties of three subspaces?

I have E, F, G as subspaces of V. Confused as to how to start proving that if E + F = E + G, then F = G. Also the same except with direct sum. Assuming it involves evaluating combined summation but ...
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1answer
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Show this representation of $\mathfrak{sl}_2(\mathbb C)$ is completely reducible

Consider the homomorphism $\phi : \mathfrak{sl}_2(\mathbb C) \rightarrow \mathfrak{sl}_3(\mathbb C)$ sending $\left(\begin{matrix} a&b \\ c&d \end{matrix}\right)$ to $\left(\begin{matrix} a&...
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Theorem about projections and direct sums

I'm in advanced linear algebra and our professor presented this theorem about projections and direct sums: -If {P1,...,Pr} is a complete set of orthogonal projections then V=V1⊕...⊕Vr ...
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56 views

Direct Sum involving integration

Let $U$ be the subspace of polynomials $P$ defined as follows $$U=\left\{p(x)\in P ~ \text{so that } \int _0^1 p\left(x\right)dx=\int _0^1xp\left(x\right)dx=0\right\}.$$ Is the following ...
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1answer
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Existence of subspace

V is a vector space over the field F. W is a proper subspace of V. Does there exist a subspace U such that V is the direct sum of U and W. I have been able to prove that such a subspace exists in the ...
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Addition and Direct Sums of Subspaces

Let $A$, $B$ and $C$ be subspaces of a vector space $V$. Prove or falsify the following: a) If $A$ $+$ $B$ $=$ $A$ $+$ $C$, then $B$ $=$ $C$. b) If $A$ $⊕$ $B$ $=$ $A$ $⊕$ $C$ then $B$ $=$ $C$. I ...
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Let $M=\Bbb Z^2$ . If $N$ and $P$ internal direct summands of $M$ , then does it follow that $N+P$ is also an internal direct summand of $M$

Let $M=\Bbb Z^2$ . If $N$ and $P$ internal direct summands of $M$ , then does it follow that $N+P$ is also an internal direct summand of $M$. My attempt : $\Bbb Z^2=\Bbb Z \oplus \Bbb Z$. Then $N=P=\...
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If $M = P \oplus Q$ then is $M\cap N = (P \cap N) \oplus (Q \cap N)$

(1) If $M,N,P,Q$ are $R$-modules and $M = P \oplus Q$ then is $M\cap N = (P \cap N) \oplus (Q \cap N)?$ Since the modules are not said to be anything like finitely generated or free, I really don't ...
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Is $\Bbb Q$ a decomposable module over $\Bbb Z$ or not?

Is $\Bbb Q$ a decomposable module over $\Bbb Z$ or not? My attempt: let, $p_1,p_2,\dots,p_k,\dots$ be an enumeration of primes in $\Bbb N$. Then, can't we write $\Bbb Q = \Bbb Z \oplus Z(\frac{1}{...
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83 views

Linear algebra: Direct sum decomposition

Let $V$, $W$ be $F$-vector spaces such that $V$ has a direct sum decomposition $V= U_1+U_2$. Let $F_1: U_1 → W$ and $F_2: U_2 → W$ be two linear maps. We say $F : V → W$ is a common extension of $F_1$ ...
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2answers
105 views

What is the difference between an infinite set of 1 dollar bills and an infinite set of 20 dollar bills?

Even though both sets approach infinity at different increments, do they eventually approach infinity at the same value at the same degree? Or is the second set of infinite 20 dollar bills 20 times ...
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1answer
48 views

Direct sum of groups categorically [duplicate]

I mean groups and not abelian groups. In Grp, categorically, he product is the cartesian product and the coproduct is the free product. So what place takes the direct sum? Can it be defined ...
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Explaining the $K$ functor isomorphism $K(X) \cong \tilde{K}(X) \oplus \Bbb Z$

It seems unclear to me what the splitting means in page 40, of Hatcher's for a $K$ functor. There is a natural homomorphism, $\varphi:K(X)\rightarrow \tilde{K}(X)$, sending $[E-\epsilon^n] \...
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1answer
44 views

Finding the projections associated to the direct sum decomposition of a vector space

PROBLEM STATEMENT Hi all. First, I am given two vector spaces: Let $V = \mathbb{R}^4$. Consider the following subspaces: $V_1 = \{(x,y,z,t)\ : x = y = z\}, V_2=[(2,1,1,1)], V_3 =[(2,2,1,1)]$ And ...
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1answer
43 views

Ideals of $R = R_1 \oplus R_2 \oplus \dots \oplus R_n$

I'm trying to show that if $A$ is an ideal of $R = R_1 \oplus R_2 \oplus \dots \oplus R_n$ then $A = A \cap R_1 \oplus A \cap R_2 \oplus \dots \oplus A \cap R_n$ I've been running around in my head ...
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2answers
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Why is a direct summand of a compact object compact?

In an additive category, we say that an object $A$ is compact if the functor $\text{Hom}(A, -)$ respects coproducts. That is, if the canonical morphism $$ \coprod_{i} \text{Hom} \left( A, X_{i} \right)...
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1answer
32 views

Injectivity and surjectivity of $\prod_{i=1}^m W_i \to \sum_{i=1}^m W_i$ given by $(w_i)_i \mapsto \sum_i w_i$

Suppose $W_1,W_2,\dotsc,W_m$ are subspaces of $V$. Define a map $$ T \colon W_1 \times W_2 \times \dotsb \times W_m \to W_1 + W_2 + \dotsb + W_m $$ by $$ T(w_1,w_2,...,w_m) = ...
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1answer
56 views

Summation with multiple variables seprated by comma.

Suppose there are two summations. 1)$\sum_{i=1}^na_i$ 2) $\sum_{i=1}^nb_i$ Both summations are same but have different variables. Instead of writing both separately, Is it possible to merge them ...