Editing Direct sum of modules (section)

== Construction for vector spaces and abelian groups ==

We give the construction first in these two cases, under the assumption that we have only two objects. Then we generalize to an arbitrary family of arbitrary modules. The key elements of the general construction are more clearly identified by considering these two cases in depth.

=== Construction for two vector spaces ===

Suppose ''V'' and ''W'' are [[vector space]]s over the [[field (mathematics)|field]] ''K''. The [[Cartesian product]] ''V'' × ''W'' can be given the structure of a vector space over ''K'' {{harv|Halmos|1974|loc=§18}} by defining the operations componentwise:

* (''v''<sub>1</sub>, ''w''<sub>1</sub>) + (''v''<sub>2</sub>, ''w''<sub>2</sub>) = (''v''<sub>1</sub> + ''v''<sub>2</sub>, ''w''<sub>1</sub> + ''w''<sub>2</sub>)
* ''α'' (''v'', ''w'') = (''α'' ''v'', ''α'' ''w'')

for ''v'', ''v''<sub>1</sub>, ''v''<sub>2</sub> ∈ ''V'', ''w'', ''w''<sub>1</sub>, ''w''<sub>2</sub> ∈ ''W'', and ''α'' ∈ ''K''.

The resulting vector space is called the ''direct sum'' of ''V'' and ''W'' and is usually denoted by a plus symbol inside a circle:
<math display=block>V \oplus W</math>

It is customary to write the elements of an ordered sum not as ordered pairs (''v'', ''w''), but as a sum ''v'' + ''w''.

The subspace ''V'' × {0} of ''V'' ⊕ ''W'' is isomorphic to ''V'' and is often identified with ''V''; similarly for {0} × ''W'' and ''W''. (See ''internal direct sum'' below.) With this identification, every element of ''V'' ⊕ ''W'' can be written in one and only one way as the sum of an element of ''V'' and an element of ''W''. The [[dimension of a vector space|dimension]] of ''V'' ⊕ ''W'' is equal to the sum of the dimensions of ''V'' and ''W''.  One elementary use is the reconstruction of a finite vector space from any subspace ''W'' and its orthogonal complement:  
<math display=block>\mathbb{R}^n = W \oplus W^{\perp}</math>

This construction readily generalizes to any [[finite set|finite]] number of vector spaces.

=== Construction for two abelian groups ===

For [[abelian group]]s ''G'' and ''H'' which are written additively, the [[direct product]] of ''G'' and ''H'' is also called a direct sum {{harv|Mac Lane|Birkhoff|1999|loc=§V.6}}. Thus the [[Cartesian product]] ''G'' × ''H'' is equipped with the structure of an abelian group by defining the operations componentwise:
: (''g''<sub>1</sub>, ''h''<sub>1</sub>) + (''g''<sub>2</sub>, ''h''<sub>2</sub>) = (''g''<sub>1</sub> + ''g''<sub>2</sub>, ''h''<sub>1</sub> + ''h''<sub>2</sub>)

for ''g''<sub>1</sub>, ''g''<sub>2</sub> in ''G'', and ''h''<sub>1</sub>, ''h''<sub>2</sub> in ''H''.

Integral multiples are similarly defined componentwise by
: ''n''(''g'', ''h'') = (''ng'', ''nh'')

for ''g'' in ''G'', ''h'' in ''H'', and ''n'' an [[integer]]. This parallels the extension of the scalar product of vector spaces to the direct sum above.

The resulting abelian group is called the ''direct sum'' of ''G'' and ''H'' and is usually denoted by a plus symbol inside a circle:
<math display=block>G \oplus H</math>

It is customary to write the elements of an ordered sum not as ordered pairs (''g'', ''h''), but as a sum ''g'' + ''h''.

The [[subgroup]] ''G'' × {0} of ''G'' ⊕ ''H'' is isomorphic to ''G'' and is often identified with ''G''; similarly for {0} × ''H'' and ''H''. (See [[Direct_sum_of_modules#Internal_direct_sum|''internal direct sum'']] below.) With this identification, it is true that every element of ''G'' ⊕ ''H'' can be written in one and only one way as the sum of an element of ''G'' and an element of ''H''. The [[rank of an abelian group|rank]] of ''G'' ⊕ ''H'' is equal to the sum of the ranks of ''G'' and ''H''.

This construction readily generalises to any [[finite set|finite]] number of abelian groups.