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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.

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Field & Integral Domain

Labels: Abstract Algebra, Field

Integral domain:-

1- A commutative ring with unity element and without zero divisors is called an integral domain.

2- An algebraic system (D,+,.) where D is a non-empty set with two binary compositions to be denoted by addition and multiplication is called an integral domain if following axioms are satisfied-

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Definition of Ring

Labels: Abstract Algebra, Ring

Ring:-

An algebraic system (R,+,.) consisting of a non-empty set R with two binary composition (to be denoted by addition and multiplication) is called a ring if following axiom are satisfied-

(R,+) is an abelian group.

1- Closure axiom

2- Associative law

3- Identity element

4- Inverse axiom

(R,.) is a semi group.

1- Closure axiom

2- Associative law

3- Multiplication distributive over addition i.e.

a.(b+c)=a.b+a.c, ∀ a, b, c∊R

and (b+c).a=b.a+c.a, ∀a, b, c∊R

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Linear Transformation Theorem-1

Labels: Linear Algebra, Linear Transformation Theorems

Statement:-

Let T be a linear transformation from avector space U into V over the field F.Then T is non-singular iff T is one-one.

Proof:-

Let T be a non-singular transformation from U into V.

Let α₁,α₂∊U such that

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Vector Space Question-1

Labels: Linear Algebra, Vector Space Questions

Question:- Express (1,2,3) as a linear combination of (1,1,1),(2,-1,1) and (1,-2,5) in V₃(R).

Solution:- Let a_1,a₂,a₃∊R such that

(1,2,3)=a₁(1,1,1)+a₂(2,-1,1)+a₃(1,-2,5)……………………(a)

(1,2,3)=(a_1,a_1,a₁)+(2a₂,-a₂,a₂)+(a₃,-2a₃,5a₃)

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Definition of Group

Labels: Abstract Algebra, Group

An algebraic system (G,o) where G be a non-empty set with o as defined binary operation is called a group if following axiom are satisfied-

1- Closure Axiom:-

If a, b∊G ⇨ aob∊G , ∀ a,b∊G

then G is said to be closed under the binary operation.

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Binary Operation & Algebric Structure

Labels: Abstract Algebra, Group

Binary operation or Binary composition on a set:-

Let G be a non-empty set then an operation ‘o’ on the non-empty set G is called binary operation.

If a∊G, b∊G ⇒ aob∊G, ∀ a, b∊G

This property is called closure property and if this is satisfied then G is said to be closed under the binary composition ‘o’.

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Bilinear Form:-

Let U and V be two vector spaces over the same field F. A bilinear form on W=U⊕V is a fraction f from W into F, which assigns to each element (α,β) in W a scalar f(α,β) in such a way that

f(aα₁+bα₂,β)= af(α₁,β)+bf(α₂,β)

& f(α,aβ₁+bβ₂)=af(α,β₁)+bf(α,β₂)

Here f(α,β) is an element of F. It denotes the image of (α,β) under the function f. Thus a bilinear form on W is a function from W into F which is linear as a function of either of its arguments when the other is fixed.

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Linear Transformation (Question-1)

Labels: Linear Algebra, Linear Transformation Questions

Question:-

Show that the mapping T:V₂(R)→V₃(R) defined by T(a,b)=(a+b, a-b, b) is a linear transformation from V₂(R) into V₃(R).

Find the range, rank, null space and null(T) of T.

Solution:- Given that

T:V₂(R)→V₃(R)

Such that T(a,b)=(a+b,a-b,b)

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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,

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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

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Null Space, Rank, Nullity & Range

Labels: Linear Algebra, Linear Tranformation

Range of a Linear Transformation:-

Let U and V be two vector spaces over the field F.Let T be a linear transformation from U into V then the of all vectors of V which are images of elements of U is called range of linear transformation. It is denoted by R(T).

Thus R(T)={T(α)∊V:α∊U}

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Annihilator Theorem-1

Labels: Annihilator, Linear Algebra

Theorem:-

Let V be a finite dimensional vector space over the field F and let W be a subspace over the field F and let W be a subspace of V.Then W⁰⁰=W.

Proof:-

We have

W⁰={f∊V’ :f(α)=0 ∀ α∊W}………..(1)

And W⁰⁰={α∊V : f(α)=0 ∀ f∊W⁰}……..(2)

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Definition of annihilator

Labels: Annihilator, Linear Algebra

Annihilators:-

If V is a vector space over the field F and S is a subset of V,the annihilator of S is the set S^o of all linear functional f on V such that

f(α)=0 ∀ α∊S

Sometimes A(S) is also used to denote the annihilators of S.

Thus S⁰={f∊V’: f(α)=0 ∀ α∊S}

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Annihilator Question-1

Labels: Annihilator, Linear Algebra

Question:-Let W₁ and W₂ be subspaces of a finite dimensional vector space V.

(a)Prove that (W₁+W₂)^o=W₁⁰⋂W₂⁰

(b)Prove that (W₁⋂W₂)⁰=W₁⁰+W₂⁰.

Solution:-

(a) First we shall prove that

W₁⋂W₂⊆(W₁+W₂)⁰.

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One-One, onto & Invertible linear transformation

Labels: Linear Algebra, Linear Tranformation

One-one transformation:-

Let T be a transformation from a vector space U into V then T is said to be one-one transformation if

α₁,α₂∊U and α₁≠α₂ ⇒ T(α₁)≠T(α₂)

In other word

α₁,α₂∊U and T(α₁)=T(α₂) ⇒ α₁=α₂

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Linear Transformation

Labels: Linear Algebra, Linear Tranformation

Let U & V be two vector spaces over the same field F.A linear transformation from U into V is a function T:U→V, such that,

T(aα+bβ)=aT(α)+bT(β),for every a,b∊F and α,β∊U

This condition is also called linearity property.

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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.

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Linear Sum of two subspaces

Labels: Linear Algebra, Vector Space

Let V be a vector space over the field F the linear sum of two subspaces W₁ and W₂ of V written as (W₁+W₂) and is defined as W₁+W₂={α₁+α₂:α₁єw₁,α₂єw₂} which shows that each element 0f (W₁+W₂) is expressible as sum of an element of W₁ and an element of W₂.

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Linear Combination & Linear Span

Labels: Linear Algebra, Vector Space

Linear Combination Of Vectors:-

Let V be a vector space over the field F,then a vector αєV is said to be a linear combination of vectors α1,α2,α3,………………..αnєV.

If α=a1α1+a2α2+a3α3+…………………………+anαn

where a1,a2,a3,……………………………….anєF

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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.

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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.

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Inner Product Space

Labels: Inner Product Space, Linear Algebra

Let V be the vector space over the field F where F is either field of real numbers or field of complex number. An inner product space on V is a function from VχV into F which assigns to each ordered pairs of vectors α,β in V by a scalar (α,β) such that

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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.

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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}

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Direct Sum of a vector space

Labels: Linear Algebra, Vector Space

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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