The direct sum of two vector spaces V and W is the set V⊕W of pairs of vectors (v, w) in V and W, with the operations
(v, w) + (v', w') = (v + v', w + w'),
c(v, w) = (cv, cw).
The direct sum of more than two vector spaces can be similarly defined.
The proof is a routine verification of the eight properties. The following verifies the first property
(v, w) + (v', w')
= (v + v', w + w') (definition)
= (v' + v, w' + w) (commutativity of additions in V and W)
= (v', w') + (v, w). (definition)
The zero vector in V⊕W is given by (0V, 0W). The nagative of (v, w) is (- v, - w). The verification of the other properties are similar.
Example We often think of a 2 by 3 matrix as columns: A = [u v w], where u, v, w ∈ R2. This gives a natural isomorphism between M(2, 3) and R2⊕R2⊕R2. Moreover, this particular example of partitioning 4 by 5 matrices gives a natural isomorphism between M(4, 5) and M(3, 3)⊕M(3, 2)⊕M(1, 3)⊕M(1, 2).