Learn Pure Mathematics

Showing posts with label Vector Space Theorem. Show all posts

Vector Space (Theorem-5)

Labels: Linear Algebra, Vector Space Theorem

Statement:- The necessary & sufficient conditions for a non-empty subset W of a vector spaceV(F) to be a subspace of V are

( 1) α∊W, β∊W ⇨ α-β∊W

(2) a∊F, α∊W ⇨ aα∊W

Proof:- Necessary Condition:-

Let V be a vector space over the field F and W be its subspace.

∴ W will be a vector space over the same field F.

(0)

Vector Space (Theorem-4)

Labels: Linear Algebra, Vector Space Theorem

Question:-

Show that a field K can be regarded as a vector space over any subfield F of K.

Solution:- Let K be a field and F be its subfield.

Since, K is a field,

Therefore,

(0)

Union of two subspaces (Theorem)

Labels: Linear Algebra, Vector Space Theorem

Statement:-

The union of two subspace of a vector space is a subspace iff one each contained.

Proof:-

Let V be a vector space over the field F.Let W₁ and W₂ be its subspaces such that either

(0)

Theorem on linear sum

Labels: Linear Algebra, Vector Space Theorem

Statement:-

The linear sum of two subspaces of a vector space is also a subspace of same vector space.

(0)

Invarience Theorem

Labels: Linear Algebra, Vector Space Theorem

Statement:- Any two bases of a finite dimensional vector space have same number of elements.

The number of elements in a basis of a finite dimensional vector space is unique.

(0)

Intersection of two subspaces

Labels: Linear Algebra, Vector Space Theorem

Statement:-

The intersection of any two sub-spaces of a vector space is also a subspace of the same vector space.

Proof:-

Let V be a vector space over the field F and W1 and W2 be its subspaces.

(0)

Existence Theorem

Labels: Linear Algebra, Vector Space Theorem

Statement:-There exists a basis for each finite dimensional vector space.

Proof:-

Let V be vector space & let S={ α₁,α₂,……………..,α_m } be a finite subset of V such that L(S)=V. If S is linearly independent then it is a basis of V.

(2)

Vector Space (Theorem-3)

Labels: Linear Algebra, Vector Space Theorem

Statement:- The necessary and sufficient condition for a vector space V over the field F to be direct sum of its two subspaces W₁ and W₂ are that

1-V=W₁+W₂

2- W₁ and W₂ are disjoint i.e. W₁∩W₂={0}

(0)

Vector Space (Theorem-2)

Labels: Linear Algebra, Vector Space Theorem

Theorem:- If W is a subspace of an n-dimensional vector space over the field F then

dimW≤dimV

Proof:-

Let W be a subspace of a finite dimensional vector space V(F).Let S={α₁,α₂,…………………..α_m} be a basis of V.

(0)

Categories

Archives

Vector Space (Theorem-5)

Statement:- The necessary & sufficient conditions for a non-empty subset W of a vector spaceV(F) to be a subspace of V are

Proof:- Necessary Condition:-

Vector Space (Theorem-4)

Question:-

Solution:- Let K be a field and F be its subfield.

Union of two subspaces (Theorem)

Statement:-

Proof:-

Theorem on linear sum

Statement:-

Invarience Theorem

Intersection of two subspaces

Existence Theorem

Vector Space (Theorem-3)

Vector Space (Theorem-2)