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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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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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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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Categories

Archives

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:-

Statement:-

Proof:-

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

Bilinear Form:-

Question:-

Solution:- Given that

Question:-

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

Statement:-

Proof:-

Range of a Linear Transformation:-

Theorem:-

Proof:-

Annihilators:-

One-one transformation:-

Statement:-

Linear Combination Of Vectors:-

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