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Questions tagged [direct-sum]

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

Direct sums are defined for a number of different sorts of mathematical objects, including subspaces, matrices, modules, and groups.

Definition: Let $~U,~ W~$ be subspaces of $~V~$ . Then $~V~$ is said to be the direct sum of $~U~$ and $~W~$, and we write $~V = U ⊕ W~$, if $~V = U + W~$ and $~U ∩ W = \{0\}~$.

  • The significant property of the direct sum is that it is the coproduct in the category of modules (i.e., a module direct sum).
  • Direct products and direct sums differ for infinite indices. An element of the direct sum is zero for all but a finite number of entries, while an element of the direct product can have all nonzero entries.
  • The direct sum of Abelian groups is the same as the group direct product, but that the term direct sum is not used for groups which are non-Abelian.

References:

https://en.wikipedia.org/wiki/Direct_sum

http://mathworld.wolfram.com/DirectSum.html

1081 questions
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Understanding Direct Sum

The definition of direct sum that I learned is: $U$ is the direct sum of $U_1,U_2$ if $\forall u\in U,\exists !(u_1,u_2)\in U_1\times U_2:u=u_1+u_2$. I want to understand what $U=U_1\oplus U_2$ means. From the definition I know that every element of…
Guilherme Salomé
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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: $M+N=\{(x,y)\in \mathbb{R}^2|(x,y)=(x_1,y_1)+(x_2,y_2);\ (x_1,y_1)\in M,\ (x_2,y_2) \in N \}$ where $x=x_1+x_2$ and…
AK Math
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$\operatorname{Im} T^t \,\dot+\, \ker T = V$

Let T be sqaure matrix and regarded as a linear operator on a finite dimensional vector space V such that $T^2 = 0$. Then is $\operatorname{Im} T^t \,\dot+\, \ker T = V$ obvious? If so, why is it so? ($\dot+$ denotes direct sum ) Addtional…
Beverlie
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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 in $G$ and $C$ is normal in $G$.
I_wil_break_you as
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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?"
Fatih
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Generalizing the formula for dim$(U+W)$

Is there a way to generalize $\dim(\sum_{i=1}^{k}W_i)$ from the following formula? $\dim(U+W)=\dim U+\dim W-\dim (U \cap W)$ I tried looking at examples $k=3$: $\dim(W_1+W_2+W_3)=\dim(W_1+W_2)+\dim W_3-\dim((W_1+W_2) \cap W_3)=$ $\dim W_1+\dim…
Theorem
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How to solve:$\sum\limits_{i=0}^{x-1} {2^i} = N$

How to solve:$\sum\limits_{i=0}^{x-1} {2^i} = N$ e.g. if N=15, then x=4.
lucky1928
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