The direct sum is an operation from abstract algebra, a branch of mathematics. For example, the direct sum
R
⊕
R
{\displaystyle \mathbf {R} \oplus \mathbf {R} }
, where
R
{\displaystyle \mathbf {R} }
is real coordinate space, is the Cartesian plane,
R
2
{\displaystyle \mathbf {R} ^{2}}
. To see how the direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups
A
{\displaystyle A}
and
B
{\displaystyle B}
is another abelian group
A
⊕
B
{\displaystyle A\oplus B}
consisting of the ordered pairs
(
a
,
b
)
{\displaystyle (a,b)}
where
a
∈
A
{\displaystyle a\in A}
and
b
∈
B
{\displaystyle b\in B}
. (Confusingly this ordered pair is also called the cartesian product of the two groups.) To add ordered pairs, we define the sum
(
a
,
b
)
+
(
c
,
d
)
{\displaystyle (a,b)+(c,d)}
to be
(
a
+
c
,
b
+
d
)
{\displaystyle (a+c,b+d)}
; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of two vector spaces or two modules.
We can also form direct sums with any finite number of summands, for example
A
⊕
B
⊕
C
{\displaystyle A\oplus B\oplus C}
, provided
A
,
B
,
{\displaystyle A,B,}
and
C
{\displaystyle C}
are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces). This relies on the fact that the direct sum is associative up to isomorphism. That is,
(
A
⊕
B
)
⊕
C
≅
A
⊕
(
B
⊕
C
)
{\displaystyle (A\oplus B)\oplus C\cong A\oplus (B\oplus C)}
for any algebraic structures
A
{\displaystyle A}
,
B
{\displaystyle B}
, and
C
{\displaystyle C}
of the same kind. The direct sum is also commutative up to isomorphism, i.e.
A
⊕
B
≅
B
⊕
A
{\displaystyle A\oplus B\cong B\oplus A}
for any algebraic structures
A
{\displaystyle A}
and
B
{\displaystyle B}
of the same kind.
In the case of two summands, or any finite number of summands, the direct sum is the same as the direct product. If the arithmetic operation is written as +, as it usually is in abelian groups, then we use the direct sum. If the arithmetic operation is written as × or ⋅ or using juxtaposition (as in the expression
x
y
{\displaystyle xy}
) we use direct product.
In the case where infinitely many objects are combined, most authors make a distinction between direct sum and direct product. As an example, consider the direct sum and direct product of infinitely many real lines. An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there would be a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both. More generally, if a + sign is used, all but finitely many coordinates must be zero, while if some form of multiplication is used, all but finitely many coordinates must be 1. In more technical language, if the summands are
(
A
i
)
i
∈
I
{\displaystyle (A_{i})_{i\in I}}
, the direct sum
⨁
i
∈
I
A
i
{\displaystyle \bigoplus _{i\in I}A_{i}}
is defined to be the set of tuples
(
a
i
)
i
∈
I
{\displaystyle (a_{i})_{i\in I}}
with
a
i
∈
A
i
{\displaystyle a_{i}\in A_{i}}
such that
a
i
=
0
{\displaystyle a_{i}=0}
for all but finitely many i. The direct sum
⨁
i
∈
I
A
i
{\displaystyle \bigoplus _{i\in I}A_{i}}
is contained in the direct product
∏
i
∈
I
A
i
{\displaystyle \prod _{i\in I}A_{i}}
, but is usually strictly smaller when the index set
I
{\displaystyle I}
is infinite, because direct products do not have the restriction that all but finitely many coordinates must be zero.
Hi, I'm struggling with understanding the idea of tensor product and direct sum beyond the very basics. I know that direct sum of 2 vectors basically stacks one on top of another - I don't understand more than this . For tensor product I know that for a product of 2 matrices A and B the tensor...
Given two subspaces ##U_1, U_2##, I understand the concept of direct sum
$$ W= U_1 \oplus U_2 \iff W= U_1 + U_2, \quad U_1 \cap U_2 = \{ 0 \}$$
Where ##W## is a subspace of ##V##.
I am trying to generalize it for more than ##2## subspaces, say ##3##. I thought of the following.
$$ W= U_1...
Let's say I want to study subalgebras of the indefinite orthogonal algebra ##\mathfrak{o}(m,n)## (corresponding to the group ##O(m,n)##, with ##m## and ##n## being some positive integers), and am told that it can be decomposed into the direct sum $$\mathfrak{o}(m,n) = \mathfrak{o}(m-x,n-x)...
I was in an earlier problem tasked to do the same but when V = ##M_{2,2}(\mathbb R)##. Then i represented each matrix in V as a vector ##(a_{11}, a_{12}, a_{21}, a_{22})## and the operation ##L(A)## could be represented as ##L(A) = (a_{11}, a_{21}, a_{12}, a_{22})##. This method doesn't really...
If one shows that ##U\cap V=\{\textbf{0}\}##, which is easily shown, would that also imply ##\mathbf{R}^3=U \bigoplus V##? Or does one need to show that ##\mathbf{R}^3=U+V##? If yes, how? By defining say ##x_1'=x_1+t,x_2'=x_2+t,x_3'=x_3+2t## and hence any ##\textbf{x}=(x_1',x_2',x_3') \in...
In "Sheldon Axler's Linear Algebra Done Right, 3rd edition", on page 21 "internal direct sum", or direct sum as the author uses, is defined as such:
Following that there is a statement, titled "Condition for a direct sum" on page 23, that specifies the condition for a sum of subspaces to be...
Homework Statement
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 let ##V = M_n(\mathbb{k})##. Consider the following subspaces:
##V_1 = \{(a_{ij}) \in V : a_{ij} = 0,\forall i < j\}##
##V_2 =...
⇒Homework Statement
[/B]
Calculate ##S + T## and determine if the sum is direct for the following subspaces of ##\mathbf R^3##
a) ## S = \{(x,y,z) \in \mathbf R^3 : x =z\}##
## T = \{(x,y,z) \in R^3 : z = 0\}##
b) ## S = \{(x,y,z) \in \mathbf R^3 : x = y\}##
## T = \{(x,y,z) \in \mathbf R^3 ...
I am new on this forum, this is my gift for you.
Suppose ##(M_i)_{i \in I}## is a family of left ##R##-modules and ##M = \bigoplus_{i \in I} M_i## (external direct sum).
Suppose ##N = \langle x_1, \cdots ,x_m \rangle## is a finitely generated submodule of ##M##.
Then for each ##j = 1, \cdots...
Suppose $(M_i)_{i \in I}$ is a family of left $R$-modules and $M = \bigoplus_{i \in I} M_i$.
Suppose $N = \langle x_1 \cdots x_m \rangle$ is a finitely generated submodule of $M$.
Then for each $j = 1 \cdots m$, there is a finite $I_j \subset I$ such that $x_j \in \bigoplus_{i \in I_j} M_i$...
Homework Statement
Suppose V=U⊕W. Prove that V0=U0⊕W0. (V0= annihilator of V).
Homework Equations
(U+W)0=U0∩W0
The Attempt at a Solution
Well, I don't see how this is possible. If V0=U0⊕W0, then U0∩W0={0}, and since (U+W)0=U0∩W0, it means (U+W)0={0}, but V=U⊕W, so V0={0}. I don't think this...
Let V be a vector space. If U 1 and U2 are subspaces of V s.t. U1+U2 = V and U1 and U1∩U2 = {0V}, then we say that V is the internal direct sum of U1 and U2. In this case we write V = U1⊕U2. Show that V is internal direct sum of U1 and U2if and only if every vector in V may be written uniquely...
(From Hoffman and Kunze, Linear Algebra: Chapter 6.7, Exercise 11.) Note that ##V_j^0## means the annihilator of the space ##V_j##. V* means the dual space of V.
1. Homework Statement
Let V be a vector space, Let ##W_1 , \cdots , W_k## be subspaces of V, and let
$$V_j = W_1 + \cdots + W_{j-1}...
Homework Statement
Homework Equations
N/A
The Attempt at a Solution
I proved the first part of the question (first quote) and got stuck in the second (second quote).
I defined Im(E1) as U and Im(E2) as W and proved that v=u+w where v ∈ V, u ∈ U and w ∈ W. After that however I got stuck at...
This has turned out to be a long question to type out so I apologise, but I don't think it's too hard to follow or read through quickly and I believe the actual question itself may not be too complicated once I get round to asking it. You can possibly skip to the last few paragraphs and still be...
Hello! I am reading something about applications of group theory in quantum mechanics and I got confused about the difference between direct sum and direct product. In many places I found that they mean the same thing. However, the ways I found them defined in the book I read from, seem to be...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
In Section 10.2 Cooperstein writes the following, essentially about external direct sums ... ...
Cooperstein asserts that properties (a) and (b) above "characterize the space ##V## as the direct sum of...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
In Section 10.2 Cooperstein writes the following, essentially about external direct sums ... ...
Cooperstein asserts that properties (a) and (b) above "characterize the space V as the direct sum of the...
During lecture, the professor gave us a theorem he wants us to prove on our own before he goes over the theorem in lecture.
Theorem: Let ##V_1, V_2, ... V_n## be subspaces of a vector space ##V##. Then the following statements are equivalent.
##W=\sum V_i## is a direct sum.
Decomposition of...
Let $x \in R - \{0\},$ where $R$ is a domain.
Define $T_x(M) = \{m \in M \ | \ x^n m=0 \ \ \mathrm{for \ some} \ n \in \mathbb{N}\}$ as the $x$-torsion of $M.$
I know that $T_x(M \oplus N) = T_x(M) \oplus T_x(N)$ for $R$-modules $M,N$ only if $R$ is a PID.
But I can't think of a...
Letting $X$ be a ring and $K$ be an $X$-module, I need to show that **if** $K \cong A \times B$ for some $X$-modules $A,B$, **then** $\exists$ submodules $M'$ and $N'$ of $K$ such that:
$K=M' \oplus N'$
$M' \cong A$
$N' \cong B.$----------I understand the concepts of internal and external...
Show the direct sum of a family of free abelian groups is a free abelian group.
My first thought was to just say that since each group is free abelian we know it has a non empty basis. Then we can take the direct sum of the basis to be the basis of the direct sum of a family of free abelian...
I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra.
I need some help with some issues regarding Theorem 2.31 (regarding the direct sum of n vector spaces) on pages 61-62.
Theorem 2.31 and its accompanying text...
I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra.
I need some help with some issues regarding the Universal Mapping Property of direct sums of vector spaces as dealt with by Knapp of pages 60-61. I am not...
The definition (taken from Robert Gilmore's: Lie groups, Lie algebras, and some of their applications):
We have two vector spaces V_1 and V_2 with bases \{e_i\} and \{f_i\}. A basis for the direct product space V_1\otimes V_2 can be taken as \{e_i\otimes f_j\}. So an element w of this space...
Hi guys, I have this general question.
If we are asked to show that the direct sum of ##U+W=V##where ##U## and ##W## are subspaces of ##V=\mathbb{R}^{n}##, would it be possible for us to do so by showing that the generators of the ##U## and ##W## span ##V##? Afterwards we show that their...
Hi everyone, :)
I encountered this question and thought about it several hours. I am writing down my answer. I would greatly appreciate if somebody could find a fault in my answer or else confirm it is correct. :)
Problem:
Let \(V_1,\,\cdots,\,V_k\) be subspaces in a vector space \(V\)...
Homework Statement
Hey everyone!
So to elaborate the title a bit more: basically I have to show that the natural representation of S_{3} is a direct sum of the one-dimensional irreducible representation and the two-dimensional irreducible representation of S_{3}.
Homework Equations
Im...
Homework Statement
Let ##T\in L(V,V)## such that ##T^{2}=1##. Show that ##V=V_{+}\oplus V_{-}## where ##V_{+}=\{v\in V:T(v)=v\}## and ##V_{-}=\{v\in V:T(v)=-v\}##.The Attempt at a Solution
I was given a theorem that said that ##V## is the direct sum if and only if every vector in ##V## can be...
Is the following statement true? I am trying to see if I can use it as a lemma for a larger proof:
Let ##V## be a vector space and let ##W, W_{1},W_{2}...W_{k} ## be subspaces of ##V##.
Suppose that ## W_{1} \bigoplus W_{2} \bigoplus ... \bigoplus W_{k} = W ##
Then is it always the case that...
Is this true? I am studying direct sums and was wondering if the following statement holds? It seems to be true if one considers the proof of the dimension theorem, but I need to be sure, so I can steer my proof toward a particular direction.
## N(T) \bigoplus R(T) = V ## where ##V## is the...
I'm curious about whether a statement I conjecture about direct sums is true.
Suppose that ##V## is a finite-dimensional vector space and ##W##,##W_{1}##,##W_{2}## are subspaces of ##V##. Let ## V = W_{1} \bigoplus W ## and ## V = W_{2} \bigoplus W ##.
Then is it the case that ## W_{1} = W_{2}...
I am following Friedberg's text and having some trouble understanding some of the theorems regarding diagonalizability. The proofs seem to skip some steps, so I guess I need to work through them a bit more slowly.
Given a linear operator ## T:V → V ##, with eigenspaces ## \{ E_{...
Hi everyone,
I'm having some trouble with the concept of the direct sum and product of representations.
Say I have two representations \rho_1 , \rho_2 of a group G on vector spaces V_1, V_2 respectively. Then I know their direct sum and their product are defined as
\rho_1 \oplus \rho_2 : G...
Homework Statement
Note: I need help with part (c).
Consider the representation P: S_3 \rightarrow GL_3 where P_{\sigma} is the permutation matrix associated to \sigma.
a) Determine the character \chi_P : S_3 \rightarrow \mathbb{C}
b) Find all the irreducible representations of S_3.
c)...
Let W1 = {A\in MnXn(R)| A = AT} and W2 = {A\in MnXn(R)| A = -AT}
Show that MnXn = W1 (+) W2
where the definition of direct sum is:
V is the direct sum of W1 and W2 in symbols:
V = W1 (+) W2 if:
V = W1 + W2 and
W1 \cap W2 = {0}
Attempt:
I figure I have to show each...
Hi all,
Say that I already know W1, W2 are both subspaces of a vector space V, W1∩W2={0}, and that dim(W1)+dim(W2)=dim(V)=n, can I thus conclude that V=W1+W2, namely V is the direct sum of W1 and W2?
I always see problems like "how many structurally distinct abelian groups of order (some large number) are there? I understand how we apply the theorem which tells us that every finite abelian group of order n is isomorphic to the direct sum of cyclic groups. We find this by looking at the...
Homework Statement
I'm reading from the first edition of Axler's Linear algebra done right. In the section on sums of vector subspaces, he states:
U = {(x,0,0) ∈ F3 | x ∈ F}
W = {(y,y,0) ∈ F3 | y ∈ F}
and
1.7 U + W = {(x,y,0) ∈ F3 | x,y ∈ F}
However, shouldn't the answer be U...
Exercise #17 in Linear Algebra done right is to prove that the dimension of the direct sum of subspaces of V is equal to the sum of the dimensions of the individual subspaces. I have been trying to figure this out for a few days now and I'm really stuck. Here's what I have got so far:
Choose...
I've been working on this Linear Algebra problem for a while: Let F be a field, V a vector space over F with basis \mathcal{B}=\{b_i\mid i\in I\}. Let S be a subspace of V, and let \{B_1, \dotsc, B_k\} be a partition of \mathcal{B}. Suppose that S\cap \langle B_i\rangle\neq \{0\} for all i...
Homework Statement
Let R be a unital commutative ring. Let M be an R-module and \varphi : M \rightarrow M a homomorphism.
To show: if \varphi \circ \varphi = \varphi then M=ker(\varphi)\oplus im(\varphi)
The Attempt at a Solution
I have already shown that M=ker(\varphi)\cap im(\varphi)...
I apologize for not having any attempted work, but I have no idea how to even begin tackling this proof.
Any direction would be greatly appreciated!
Mike
Homework Statement
Let V be a vector space,
Let W1, ..., Wk be subspaces of V, and,
Let Vj = W1 + ... + Wj-1 + Wj+1 + ...
For 2 vector spaces an orthogonal direct sum is a cartesian product of the spaces (with some other stuff) (http://planetmath.org/encyclopedia/OrthogonalSum.html ), and this orthogonal direct sum uses the symbol, \oplus.
However, there's an orthogonal decomposition theorem...
Hi there. I'm a long time reader, first time poster. I'm an undergraduate in Math and Economics and I am having trouble in Linear Algebra. This is the first class I have had that focuses solely on proofs, so I am in new territory.
Homework Statement
note Although the question doesn't state...
For arbitrary natural numbers a and b, I don't think the direct sum of Z_a and Z_b (considered as additive groups) is isomorphic to Z_ab. But I think if p and q are distinct primes, the direct sum of Z_p^m and Z_q^n is always isomorphic to Z_(p^m * q^n). Am I right? I've been freely using...