What is Direct sum: Definition and 86 Discussions

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.

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  1. D

    I Understanding tensor product and direct sum

    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...
  2. JD_PM

    I Understanding the concept of direct sum

    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...
  3. Rabindranath

    I Meaning of terms in a direct sum decomposition of an algebra

    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)...
  4. K

    Show that V is an internal direct sum of the eigenspaces

    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...
  5. S

    I Showing direct sum of subspaces equals vector space

    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...
  6. Calculuser

    I Confusion about the Direct Sum of Subspaces

    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...
  7. I

    Projections and direct sum

    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 =...
  8. I

    [Linear Algebra] Help with Linear Transformation exercises

    Homework Statement 1. (a) Prove that the following is a linear transformation: ##\text{T} : \mathbb k[X]_n \rightarrow \mathbb k[X]_{n+1}## ##\text{T}(a_0 + a_1X + \ldots + a_nX^n) = a_0X + \frac{a_1}{2}X^2 + \ldots + \frac{a_n}{n+1}## ##\text{Find}## ##\text{Ker}(T)## and ##\text{Im}(T)##...
  9. I

    [Linear Algebra] Linear Transformations, Kernels and Ranges

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  10. I

    [Linear Algebra] Sum & Direct Sum of Subspaces

    ⇒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 ...
  11. S

    A On a finitely generated submodule of a direct sum of modules....

    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...
  12. S

    MHB On a finiteley generated submodule of a direct sum of left R-modules

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  13. Adgorn

    Annihilator of a Direct Sum: Proving V0=U0⊕W0 for V=U⊕W

    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...
  14. Austin Chang

    I Prove that V is the internal direct sum of two subspaces

    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...
  15. VrhoZna

    Proof regarding direct sum of the dual space of a v-space

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  16. Adgorn

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  17. Anchovy

    A SU(5), 'Standard Model decomposition', direct sum etc.

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  18. S

    I Difference between direct sum and direct product

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  19. Math Amateur

    I Characterization of External Direct Sum - Cooperstein

    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...
  20. Math Amateur

    MHB Characterization of External Direct Sum - Cooperstein, pages 359 - 360

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  21. K

    Direct Sum: Vector Spaces

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  22. K

    MHB Direct sum of p-primary components of an R-module counterexample?

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  23. K

    MHB Proving that a module can be decomposed as a direct sum of submodules

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  24. J

    Are \bigoplus and \times interchangeable in direct sum and direct product?

    Under what conditions are the symbols \bigoplus and \times intechangangable?
  25. M

    MHB Direct sum of free abelian groups

    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...
  26. Math Amateur

    MHB Direct Sum of n Vector Spaces Over F - Knapp Proposition 2.31 - Pages 61-62

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  27. Math Amateur

    MHB Universal Mapping Property of a Direct Sum - Knapp Pages 60-61

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  28. T

    Direct Sum and Direct Product: Understanding the Differences in Vector Spaces

    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...
  29. Seydlitz

    Showing that V is a direct sum of two subspaces

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  30. F

    MHB Answer: Image Direct Sum & Linear Operator: Is Union Equal?

    Given 2 subspaces and a linear operator, is the image of the direct sum of the subspaces equal to the union of the images under the operator? Thanks
  31. Sudharaka

    MHB Direct Sum Property: Proving Uniqueness

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  32. D

    Show that the natural representation of S3 is a direct sum of irreps

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  33. D

    Is V a Direct Sum of V+ and V-?

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  34. B

    Algebraic properites of the direct sum

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  35. B

    Direct sum of nullspace and range

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  36. B

    Direct sum complement is unique

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  37. B

    Direct sum of the eigenspaces equals V?

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  38. Y

    Direct sum and product representation

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  39. N

    Decompose the permutation into the direct sum of irreducible reps.

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  40. T

    Showing a set of matrices is a direct sum.

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  41. L

    Showing V is the direct sum of W1 and W2

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  42. D

    Finding which direct sum of cyclic groups Z*n is isomorphic to

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  43. S

    Basic linear algebra direct sum questions

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  44. J

    Dimension of direct sum axler

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  45. I

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  46. R

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  47. J

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  48. L

    Confusion between orthogonal sum and orthogonal direct sum

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  49. N

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  50. L

    Is the direct sum of cyclic p-groups a cyclic group?

    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...
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