### Direct Sum

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

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`) = (`c`**v**, `c`**w**).

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 `V`⊕`W` is given by (**0**_{V}, **0**_{W}). 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 **R**^{5} is isomorphic to **R**^{3}⊕**R**^{2}.

Example We often think of a 2 by 3 matrix as columns: `A` = [`u v w`], where `u`, `v`, `w` ∈ **R**^{2}. This gives a natural isomorphism between `M`(2, 3) and **R**^{2}⊕**R**^{2}⊕**R**^{2}. 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).