Basis of
a vector space:-
A
non-empty subset S of a vector space V(F) is said to be its basis if
1-S is
linearly independent.
2-S
generates V i.e. L(S)=V
i.e. each
vector in V is expressible as a linear combination of element of S.
Dimension of a vector space:-
The
number of distinct element in a basis is called dimension of vector space.
1-If a
basis of a vector space is a finite set containing n-elements then V is said to
be an n-dimensional vector space and we write
dim V=n
2- A
vector space V is said to be finite-dimensional if there exist a finite subset
S of V such that
L(S)=V
3- A
vector space which is not finitely generated is known as an infinite
dimensional vector space.
Cosets:-
Let W be
any subspace of a vector space V over the field F.Let α be any element of V
then the set {α+λ: λ∊W } is called left coset of W in V
generated by α. This set is denoted by α+W.
Thus
α+W={ α+λ : λ∊W } (Left coset )
Similarly,
the set
W+α={ α+λ : λ∊W } is right coset of W in V
generated by α.
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