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}
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