Basis Dimension & Cosets Of A Vector Space

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.