# Questions tagged [direct-sum]

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

521 questions
26 views

### 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 ...
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 ...
14 views

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 ...
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. (...
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 ...
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 ...
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 ...
79 views

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 ...
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 ...
41 views

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 $M/A \cong B$.
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
### 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)\...