# 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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38. ### Direct sum and product representation

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39. ### Decompose the permutation into the direct sum of irreducible reps.

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40. ### Showing a set of matrices is a direct sum.

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41. ### Showing V is the direct sum of W1 and W2

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

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43. ### Basic linear algebra direct sum questions

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44. ### How Do You Prove the Dimension of a Direct Sum Equals the Sum of Dimensions?

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45. ### Is a subspace the direct sum of all its intersections with a partition of the basis?

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...
46. ### To show a module M = direct sum of Image and kernal of automorphism

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)...
47. ### Prove dual space has the direct sum decomposition

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 + ...
48. ### Confusion between orthogonal sum and orthogonal direct sum

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...
49. ### Proof that a vector space W is the direct sum of Ker L and Im L

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

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