# Questions tagged [direct-sum]

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

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### Find subspace $T$ of $\mathbb{R}^{3}$ such that $\mathbb{R}^{3} =V \oplus T$

\begin{array}{l} V=\{( a+2b,2a+8b+2c,a+10b+4c) \ |\ a,b,c\in \mathbb{R}\}\\ =\{a( 1,2,1) +b( 2,8,10) +c( 0,2,4) \ |\ a,b,c\in \mathbb{R}\}\\ =Sp\{( 1,2,1) ,( 2,8,10) ,( 0,2,4)\} \end{array} Then I ...
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### Why Do Additive Categories Need Zero Objects? - Motivation

At the moment, I'm trying to develop intuition behind the derived construction of what is an additive category from the (standard) category definition. (i) It seems natural that a category with the ...
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### Matrix representation of direct sum of linear operators

Given two vector spaces $V,W$ over $K$, let $F\in L(V), T\in L(W).$ If $B,U$ are ordered basis for $V,W,$ respectively, show that the matrix $\left[F\right]_B\bigoplus \left[T\right]_U$ represents ...
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### Can I derive vector space-equivalent from direct sum?

Let $V$ be a vector space and $A_1, A_2, B_1, B_2$ are subspace of $V$ such that $$A_1 \leq B_1, \ A_2 \leq B_2,\ A_1\oplus A_2 = B_1 \oplus B_2$$ Then $A_1 = B_1 , A_2 = B_2$? What I found : Due to ...
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### Show that two summands of a direct sum are invariant under a linear operator

Given two linear operators $T,S\in L(V)$ such that $T$ and $S$ commute ($T\circ S=S\circ T$) and $T^2=T$. Show that: (a) $V=ker(T)\bigoplus im(T)$. (b) each summand is invariant under $S$. So far I ...
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### The projection of the polynomial onto U parallel to V

Prove that the space $P_3$ of polynomials of degree no higher than $3$ is the direct sum of subspaces $U$ and $V$ and find the projection of the polynomial $t^3$ onto $U$ parallel to $V$, where \begin{...
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Is this true: Let $R$ be a ring with a $1$. Let $(N_{i})_{i \in I}$ be a collection of $R$-submodules of the $R$-module $M$. If $$\sum_{i \in I}N_{i} \cong \bigoplus_{i \in I}N_{i}$$ then the map $f: \... 0 votes 1 answer 57 views ### A problem about diagonalize invariant subspaces [closed] Let$V$be a non-zero finite-dimensional vector space, A belongs to End($V$). Also, for any invariant subspace$M$of A, there exists an invariant subspace$N$of A such that$V=M\oplus N$. Prove: A ... • 1 0 votes 1 answer 34 views ### If$X=M\oplus N$, is it true$(A\cap M)\oplus (A\cap N)\subseteq A$? Let$X$be a norm space and$X=M\oplus N$where$M,N$are closed subspace of$X$. Take a closed set$A\subseteq X$. I know that$(A\cap M)\oplus (A\cap N)\neq A$, for example consider$X=\mathbb{R}^2$,... • 1,297 0 votes 0 answers 44 views ### Generating sets of an Abelian group decomposed into a direct sum I’m not sure about the correctness of my assumption about the generating sets of decomposable Abelian groups, so I will be very grateful if you can tell me whether I’m right or wrong, and if I’m wrong,... 1 vote 0 answers 56 views ### Proving existence and uniqueness of direct sum of representations of unital$C^*$-algebras From Sunder's Functional Analysis: Spectral Theory, Exercise 3.4.3. Let$\{ \pi_i : A \to \mathcal{L}(\mathcal{H}_i) \}_{i\in I}$be an arbitrary family of representations of (an arbitrary unital$C^*...
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Today I decide to study some topics of Algebra and then faced up with the definition of the direct sum of modules. To let us on the same page, the definition that I'm talking about is Given a ring $R$...
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### Decomposition of $K$-finite functions

It is a theorem that a representation $(\pi,V_\pi)$ of a compact group $K$ decomposes as a direct sum of irreducible representations. My question is about Deitmar's treatment (Automorphic Forms page ...
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### Recover direct summands in derived category?

Let $E,F\in D^b(X)$, where $D^b(X)$ denotes the derived category of coherent sheaves on some smooth variety $X$. I am thinking about the following question: If $E \oplus E[1] \simeq F \oplus F[1]$, ...
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### Dimension of tensor product vs. dimension of direct product

I am really sorry if this is trivial but I come from Physics and it is really confusing for me to understand what is going on and looking at the answers on both Physics and Math SE sent me through a ...
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### Give a non-zero $R$-module $M$ such that $M \oplus M \cong M$ [duplicate]

For some reason this hasn't been asked before and I can't seem to find an example. Here $R$ is a commutative ring. Find a non-zero $R$-module $M$ such that $M \oplus M \cong M$
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### How are "types" defined in this module?

Note: This might end up being a question about a simple concept that I forgot about (I am very tired at the time of writing this, after all), so maybe try skipping to the bottom. I'm learning about ...
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### Trouble understanding direct sum in smith normal form proof

I'm struggling to understand this step in this proof in Rotman's advanced modern algebra. The related theorems are: Let R be a euclidean ring, let F be a finitely generated free R-module, and let S ...
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### Direct sum of Hilbert/Sobolev Space?

Our professor the other day wrote the following on the blackboard: "Let $u(x)=(v(x),w(x))$ be such that $u \in H^{2,1}$ where $H^{2,1}=H^2 \oplus H^1$." He then mumbled something about this ...
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### Let $V$ be a finite dimensional vector space and $W$ be a subspace

Let $V$ be a Finite Dimensional Vector Space and $W$ be a non trivial proper subspace of $V$ then the linear span $\langle V- W\rangle$ =$V$? The statement is true, & the issue here I'm facing ...
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### How are submodules characterized with respect to the isomorphism of the direct sum of multiple modules?

Consider two rings $R_1$ and $R_2$ each with their own identities, and their direct product $R_1 \times R_2$. It is well-known that the ideals of $R_1 \times R_2$ are all of the form $I_1 \times I_2$, ...
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### Necessary and sufficient condition for a $Z \oplus Z$ base

Prove that the elements $b_1=(x_1,y_1)$, $b_2=(x_2,y_2)$ form a base of $Z \oplus Z$ if and only if, $x_1y_2-y_1x_2=±1$. This exercise is from ANEIS E MODULOS ,Francisco Cesar Polcino Milles. ...
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### submodule of completely reducible module is completely reducible

I attempted to solve problems from the textbook 'Basic Abstract Algebra' by P.B. Bhattacharya, S.K. Jain, and S.R. Nagpaul, specifically Chapter 14, Section 4, but encountered difficulties in ...
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The lamplighter group can be defined by the semidirect product: $$L_2=(\mathbb{Z} _2) \wr \mathbb{Z} \cong \bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_{2} \rtimes_\phi\mathbb{Z},$$ where $\phi(1)$ &...