Editing Direct sum of modules (section)

=== 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.