### Direct Sum

##### 1. Direct sum of vector spaces

The direct sum of two vector spaces V and W is the set VW 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 direct sum of two vector spaces is a vector space.

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 VW is given by (0V, 0W). The nagative of (v, w) is (- v, - w). The verification of the other properties are similar.

Example By grouping the first three and the last two coordinates, the euclidean space R5 is isomorphic to R3R2.

Example We often think of a 2 by 3 matrix as columns: A = [u v w], where u, v, wR2. This gives a natural isomorphism between M(2, 3) and R2R2R2. 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).