Direct Sum & Disjoint Subspace

Direct sum of a vector subspace:-

 Let V be a vector space over the field F then vector space V is said to be direct sum of its subspaces W₁ and W₂ written as   V=W1⊕W₂. If each element of V is uniquely expressible as sum of an element of W1 and an element of W2.

In this case W1 and W2 are called complementary subspaces.This definition can be extended for more than two subspaces.

i.e.       vector space V is said to be direct sum of its subspaces W1,W2,W3,………………………………….,Wn if every element αєV can be written in one and only one way

α=α1+α2+α3+…………………………..+αn

 where  α1єW1,α2єW2,α3єW3,…………………….αnєWn

Disjoint subspaces:-

Two subspaces W1 and W2 of a vector space V over the field F are said to be disjoint if their intersection(∩) with zero subspace.           

i.e.  W1 andW2 are disjoint if W1∩W2={0}