LINEAR ALGEBRA - mathsbscsrtmun.files.wordpress.com€¦ · ॥॥॥ ीह ïर ° °ीीहह ï ïरर °ीह ïर: :: : ॥॥॥॥ Paper No. 10 (B.A.) / 14(B.Sc.)

Paper No. 10 (B.A.) / 14(B.Sc.) Paper No. 10 (B.A.) / 14(B.Sc.) Linear Algebra Linear Algebra ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–5th Sem. B.A./B.Sc.(Mathematics)–3rd Year–5th Sem.

॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

LINEAR ALGEBRA

Dr.N.A.Pande Associate Professor

Department of Mathematics & Statistics, Yeshwant Mahavidyalaya, Nanded – 431602

Maharashtra, INDIA



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Paper Details

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Paper Details • University : Swami Ramanand Teerth

Marathwada University, Nanded, India

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• Course : B.A./B.Sc.

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• Subject : Mathematics

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• Year : 3rd

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• Year : 3rd

• Semester : 5th

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• Year : 3rd

• Semester : 5th

• Paper No. : 10(B.A.) / 14(B.Sc.)

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• Year : 3rd

• Semester : 5th

• Paper No. : 10(B.A.) / 14(B.Sc.)

• Syllabus Effective From : 2015-16

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• Year : 3rd

• Semester : 5th

• Paper No. : 10(B.A.) / 14(B.Sc.)


• Paper Code : MT 302

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• Year : 3rd

• Semester : 5th

• Paper No. : 10(B.A.) / 14(B.Sc.)


• Paper Code : MT 302

• Marks : 40 (University) + 10 (Internal) = 50

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Syllabus

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Syllabus • Unit-I

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Syllabus • Unit-I

�Vector Spaces : Elementary Basic Concepts of Vector Spaces, Linear Independence and Bases, Dual Spaces.

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Syllabus • Unit-I


• Unit-II

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Syllabus • Unit-I


• Unit-II

� Inner Product Spaces, Fields : Extension Fields (Definitions Only).

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Syllabus • Unit-I


• Unit-II


• Unit-III

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Syllabus • Unit-I


• Unit-II


• Unit-III

�Linear Transformation : The Algebra of Linear Transformations, Characteristic Roots, Matrices.

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Text Book & Scope

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Text Book & Scope • Recommended Text Book :

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�Title : Topics in Algebra

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�Author : I. N. Herstein

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�Edition : Second Edition

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�Publisher : John Wiley & Sons

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� ISSN : 978-0-471-01090-6

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� ISSN : 978-0-471-01090-6

• Scope :

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� ISSN : 978-0-471-01090-6

• Scope :

�Unit–I : Chapter 4 : Article 4.1, 4.2, 4.3

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� ISSN : 978-0-471-01090-6

• Scope :


�Unit–II : Chapter 4 : Article 4.4

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� ISSN : 978-0-471-01090-6

• Scope :



Chapter 5 : Article 5.1 (Definitions)

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� ISSN : 978-0-471-01090-6

• Scope :



Chapter 5 : Article 5.1 (Definitions)

�Unit–III : Chapter 6 : Article 6.1, 6.2, 6.3

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Reference Books

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Reference Books • A First Course in Abstract Algebra, By J.B. Fraleigh,

Narosa Publications

• Contemporary Abstract Algebra, By Joseph Gallion, Narosa Publications

• Linear Algebra for Undergraduates, By S.R.Mangalgiri and D.K.Daftari

• First Course in Abstract Algebra, By P.B.Bhattacharya, S.K.Jain and S.R.Nagpaul

• An Introduction to Linear Algebra , By V. Krishnamurty, V. P. Mainru, J. L. Arrora

• Linear Algebra, by L. Smith, Springer-Verlag New York.

• Matrix and Linear Algebra, by K. B. Datta, Prentice Hall of India Pvt. Ltd, New Delhi, 2000.

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Vector Spaces : Basic Concepts

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Vector Spaces : Basic Concepts • Vector Space : A non-empty set V is said to be

vector space over a field F ⇔ V is abelian group under an operation denoted by + and ∀ α ∈ F, v ∈ V, there exists an element αv in V satisfying :

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�α(u + v) = αu + αv

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�α(u + v) = αu + αv

� (α + β)v = αv + βv

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�α(u + v) = αu + αv

� (α + β)v = αv + βv

� (αβ)v = α(βv)

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�α(u + v) = αu + αv

� (α + β)v = αv + βv


�1v = v

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�α(u + v) = αu + αv

� (α + β)v = αv + βv


�1v = v

• There are 9 conditions in the definition of a vector space, 5 of abelian group and 4 listed explicitly above.

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Vector Spaces : Basic Concepts • Members of vector space are called as vectors

and are denoted by smallcase Latin letters.

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• Members of field are called as scalars and are denoted by smallcase Greek letters.

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• Examples of Vector Spaces :

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�Every field is a vector space over any of its subfields

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�Every field is a vector space over any of its subfields

� If F is a field, then the set F(n) of all ordered n-tuples of members of F is a vector space over F, with respect to coordinatewise addition and coordinatewise scalar multiplication.

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Vector Spaces : Basic Concepts • Examples of Vector Spaces (Continued) :

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� If F is a field, then the set F[x] of all polynomials in x with coefficients in F is a vector space over F, with respect to coefficientwise addition and coefficientwise scalar multiplication.

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� If F is a field, then the set F[x] of all polynomials in x with coefficients in F is a vector space over F, with respect to coefficientwise addition and coefficientwise scalar multiplication.

� If F is a field, then the set Fn[x] of all polynomials of degree at most n − 1 in x with coefficients in F is a vector space over F, with respect to coefficientwise addition and coefficientwise scalar multiplication.

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Vector Spaces : Basic Concepts • A non-empty subset W of a vector space V

over F is said to be subspace of V, if W itself is vector space over F with respect to same operations in V.

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• Subspace Criteria : A non-empty subset W of a vector space V over F is said to be subspace of V ⇔ ∀ w1, w2 ∈ W, ∀ α, β ∈ F, αw1 + βw2 ∈ W.

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• Intersection of two subspaces is a subspace.

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• Intersection of two subspaces is a subspace.

• Addition of two subspaces is a subspace.

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Vector Spaces : Basic Concepts • If U and V are two vector spaces over a

common field F, then a mapping T : U → V is said to be homomorphism ⇔ ∀ u1, u2 ∈ U, ∀ α ∈ F,

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� (u1 + u2)T = u1T + u2T

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� (u1 + u2)T = u1T + u2T

� (αu1)T = α(u1T)

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� (u1 + u2)T = u1T + u2T

� (αu1)T = α(u1T)

• A one-to-one homomorphism is called isomorphism.

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� (u1 + u2)T = u1T + u2T

� (αu1)T = α(u1T)

• A one-to-one homomorphism is called isomorphism.

• Two vector spaces are isomorphic ⇔ there exists an isomorphism of one onto another.

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Vector Spaces : Basic Concepts • ‘is isomorphic to’ is an equivalence relation on

the class of all vector spaces.

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• If T : U → V is a homomorphism, then kernel of homomorphism is defined as K = {u ∈ U | uT = 0}.

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• Kernel of homomorphism is subspace of domain vector space.

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• Kernel of homomorphism is subspace of domain vector space.

• The set of all homomorphism from a vector space U to a vector space V is denoted by Hom(U, V).

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Vector Spaces : Basic Concepts • If V is a vector space over a field F, then ∀

v ∈ V, ∀ α ∈ F,

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v ∈ V, ∀ α ∈ F,

�α0 = 0

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v ∈ V, ∀ α ∈ F,

�α0 = 0

�0v = 0

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v ∈ V, ∀ α ∈ F,

�α0 = 0

�0v = 0

� (−α)v = −(αv)

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v ∈ V, ∀ α ∈ F,

�α0 = 0

�0v = 0

� (−α)v = −(αv)

�αv = 0 ⇒ α = 0 or v = 0

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v ∈ V, ∀ α ∈ F,

�α0 = 0

�0v = 0

� (−α)v = −(αv)

�αv = 0 ⇒ α = 0 or v = 0

• If V is a vector space over a field F and W is a subspace of V, then V/W = {v + W | v ∈ V}is vector space over F with respect to coset addition and scalar multiplication and is called quotient space.

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Vector Spaces : Basic Concepts • If T is a homomorphism of U onto V with

kernel W, then U/W is isomorphic to V.

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• A vector space V over F is said to be internal

direct sum of its subspaces U1, U2, ⋯⋯⋯⋯ , Un ⇔ every vector in V can be expressed as sum of members of Ui in a unique way.

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• A vector space V over F is said to be internal

direct sum of its subspaces U1, U2, ⋯⋯⋯⋯ , Un ⇔ every vector in V can be expressed as sum of members of Ui in a unique way.

• If V1, V2, ⋯⋯⋯⋯ , Vn are vector spaces over a common field F, then their Cartesian product V1 × V2 × ⋯⋯⋯⋯ × Vn is a vector space over F with respect to coordinatewise addition and coordinatewise scalar multiplication and is called external direct sum of V1, V2, ⋯⋯⋯⋯ , Vn.

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Vector Spaces : Basic Concepts • If a vector space V over F is internal direct

sum of its subspaces U1, U2, ⋯⋯⋯⋯ , Un ⇒ V is isomorphic to external direct sum of U1, U2, ⋯⋯⋯⋯ , Un.

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• In a vector space V over F, v, w ∈ V, ∀ α ∈ F, α(v − w) = αv − αw.

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• In a vector space V over F, v, w ∈ V, ∀ α ∈ F, α(v − w) = αv − αw.

• If A and B are subspaces of a vector space V over F, then (A + B)/B is isomorphic to A/(A ∩ B).

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Linear Independence and Bases

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Linear Independence and Bases • If V is a vector space over F and if

v1, v2, ⋯⋯⋯⋯ , vn are in V, then their linear combination is a vector of form α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn, where αi ∈ F.

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Linear Independence and Bases • If V is a vector space over F and if

v1, v2, ⋯⋯⋯⋯ , vn are in V, then their linear combination is a vector of form α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn, where αi ∈ F.

• If S is a non-empty subset of a vector space V over F, then linear span of S, denoted by L(S) is the set of all linear combinations of finite sets of elements of S.

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Linear Independence and Bases • Properties of L(S) :

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�L(S) is a subspace of V.

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�S ⊂ T ⇒ L(S) ⊂ L(T)

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�S ⊂ T ⇒ L(S) ⊂ L(T)

�L(S ∪ T) = L(S) + L(T)

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�S ⊂ T ⇒ L(S) ⊂ L(T)

�L(S ∪ T) = L(S) + L(T)

�L(L(S)) = L(S)

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�S ⊂ T ⇒ L(S) ⊂ L(T)

�L(S ∪ T) = L(S) + L(T)

�L(L(S)) = L(S)

• A vector space is said to be finite dimensional ⇔ ∃ a finite subset S of V such that L(S) = V.

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�S ⊂ T ⇒ L(S) ⊂ L(T)

�L(S ∪ T) = L(S) + L(T)

�L(L(S)) = L(S)

• A vector space is said to be finite dimensional ⇔ ∃ a finite subset S of V such that L(S) = V.

• A vector space is said to be infinite dimensional ⇔ it is not finite dimensional.

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Linear Independence and Bases • In a vector space V over F, v1, v2, ⋯⋯⋯⋯ , vn in V

are said to be linearly dependent ⇔ ∃ scalars α1, α2, ⋯⋯⋯⋯ , αn in F, not all zero such that α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn = 0.

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17



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• In a vector space V over F, v1, v2, ⋯⋯⋯⋯ , vn in V are said to be linearly independent ⇔ whenever α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn = 0, each αi = 0.

UNIT-I

17



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• In a vector space V over F, v1, v2, ⋯⋯⋯⋯ , vn in V are said to be linearly independent ⇔ whenever α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn = 0, each αi = 0.

• Linear dependence or independence of vectors depends upon vectors as well as field over which V is the vector space.

UNIT-I

17



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Linear Independence and Bases • v1, v2, ⋯⋯⋯⋯ , vn in V are linearly independent ⇒

every vector in their linear span as unique representation of the form α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn.

UNIT-I

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• v1, v2, ⋯⋯⋯⋯ , vn in V are linearly dependent ⇒ some vk is linear combination of preceding ones.

UNIT-I

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• v1, v2, ⋯⋯⋯⋯ , vn in V are linearly dependent ⇒ some vk is linear combination of preceding ones.

• v1, v2, ⋯⋯⋯⋯ , vn in V are linearly dependent with linear span W ⇒ there exists a subset of it which is linearly independent whose linear span is still W.

UNIT-I

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

19



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Linear Independence and Bases • A subset S of a vector space is called basis of

V ⇔ S is linearly independent and L(S) = V.

UNIT-I

19



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Basis is the largest linearly independent set.

UNIT-I

19



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• If V is a finite-dimensional vector space over F, then any two bases of V have the same number of elements.

UNIT-I

19



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• F(n) is isomorphic to F(m) ⇔ m = n.

UNIT-I

19



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• F(n) is isomorphic to F(m) ⇔ m = n.

• Each finite-dimensional vector space is isomorphic to a unique F(n), for which n is called dimension of V and is denoted by dim V.

UNIT-I

19



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

20



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Linear Independence and Bases • Any two finite dimension vector spaces having

same dimension are isomorphic.

UNIT-I

20



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Any two vector isomorphic vector spaces have same dimension.

UNIT-I

20



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• If W is a subspace of a finite-dimensional vector space V over F, then dim W ≤ dim V and dim V/W = dim V − dim W.

UNIT-I

20



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• If W is a subspace of a finite-dimensional vector space V over F, then dim W ≤ dim V and dim V/W = dim V − dim W.

• If A and B are finite-dimensional subspaces of a vector space V over F, then dim(A + B) = dim A + dim B −dim(A ∩ B).

UNIT-I

20



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Dual Spaces

UNIT-I

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Dual Spaces • If V and W are vector spaces over same field

F, then the set of all homomorphisms from V to W is denoted by Hom(V, W) and is itself a vector space over F.

UNIT-I

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• dim Hom(V, W) = dim V × dim W

UNIT-I

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• The dimension of a field F as a vector space over itself is 1.

UNIT-I

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• dim Hom(V, V) = dim V2

UNIT-I

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• dim Hom(V, V) = dim V2

• dim Hom(V, F) = dim V

UNIT-I

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Dual Spaces

UNIT-I

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Dual Spaces • If V is a vector space over F, then the vector

space Hom(V, F) is called dual space of V and is denoted by v.̂

UNIT-I

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Members of v ̂are scalar valued functions of vectors.

UNIT-I

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• dim v ̂= dim V.

UNIT-I

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• Members of v ̂are called functionals.

UNIT-I

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥






• If V is finite dimensional vector space over F, then ∀ 0 ≠ v ∈ V, ∃ f ∈ v ̂such that f(v) ≠ 0.

UNIT-I

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥






• If V is finite dimensional vector space over F, then ∀ 0 ≠ v ∈ V, ∃ f ∈ v ̂such that f(v) ≠ 0.

• Every finite-dimensional vector space V is isomorphic to its dual space v.̂

UNIT-I

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Dual Spaces

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Dual Spaces • Annihilator A(W) of a subspace W of a

vector space V is A(W) = {f ∈ v ̂| f(w) = 0 ∀ w ∈ W}

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Annihilator of a subspace is itself a subspace.

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• dim A(W) = dim V − dim W.

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• If V is a vector space over F and U & W are subspaces of V such that U ⊂ W, then A(U) ⊃ A(W).

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥






• If W is a subspace of a vector space V over F, then A(A(W)) = W.

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥






• If W is a subspace of a vector space V over F, then A(A(W)) = W.

• For any subset S of V, then A(S) = A(L(S)).

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inner Product Spaces

UNIT-II

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inner Product Spaces • An inner product space is a vector space V

over F in which there is defined a function of two vectors giving a scalar denoted by (u, v) ∈ F ∀ u, v ∈ V such that ∀ u, v, w ∈ V and ∀ α, β ∈ F,

UNIT-II

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



� (u, v) =

UNIT-II

24

),( uv



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



� (u, v) =

� (u, u) ≥ 0 ∀ u ∈ V and (u, u) = 0 ⇔ u = 0

UNIT-II

24

),( uv



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



� (u, v) =

� (u, u) ≥ 0 ∀ u ∈ V and (u, u) = 0 ⇔ u = 0

� (αu + βv, w) = α(u, w) + β(v, w)

UNIT-II

24

),( uv



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



� (u, v) =

� (u, u) ≥ 0 ∀ u ∈ V and (u, u) = 0 ⇔ u = 0

� (αu + βv, w) = α(u, w) + β(v, w)

• In an inner product space V over F, for all scalars and vectors,

UNIT-II

24

),( uv



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



� (u, v) =

� (u, u) ≥ 0 ∀ u ∈ V and (u, u) = 0 ⇔ u = 0

� (αu + βv, w) = α(u, w) + β(v, w)

• In an inner product space V over F, for all scalars and vectors,

UNIT-II

24

),( uv

),(+),(=)+,( wuvuwvu βαβα



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

25



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inner Product Spaces • Length or norm of a vector v is given by

UNIT-II

25

( )vvv ,=||||



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• || αv || = |α|⋅⋅⋅⋅||v||

UNIT-II

25

( )vvv ,=||||



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• || αv || = |α|⋅⋅⋅⋅||v||

• If a, b, c are real numbers such that aλ2 + 2bλ + c ≥ 0 ∀ real λ, then b2 ≤ ac.

UNIT-II

25

( )vvv ,=||||



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• || αv || = |α|⋅⋅⋅⋅||v||


• Schwarz’s Inequality : |(u, v)| ≤ ||u||⋅⋅⋅⋅||v||

UNIT-II

25

( )vvv ,=||||



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• || αv || = |α|⋅⋅⋅⋅||v||



• u is said to be orthogonal to v ⇔ (u, v) = 0.

UNIT-II

25

( )vvv ,=||||



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• || αv || = |α|⋅⋅⋅⋅||v||



• u is said to be orthogonal to v ⇔ (u, v) = 0.

• Orthogonal complement of a subspace W of an inner product space V is given by W⊥ = {v ∈ V | (v, w) = 0 ∀ w ∈ W}

UNIT-II

25

( )vvv ,=||||



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inner Product Spaces • W⊥ is a subspace of V.

UNIT-II

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• W ∩ W⊥ = {0}

UNIT-II

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• W ∩ W⊥ = {0}

• A set of vectors {vi} is said to be orthonormal ⇔

UNIT-II

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• W ∩ W⊥ = {0}


� (vi, vi) = 1 ∀ I all vectors are of unit length)

UNIT-II

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• W ∩ W⊥ = {0}



� (vi, vj) = 0 ∀ i ≠ j (distinct vectors are mutually⊥)

UNIT-II

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• W ∩ W⊥ = {0}




• An orthonormal set is linearly independent.

UNIT-II

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• W ∩ W⊥ = {0}




• An orthonormal set is linearly independent.

• {vi} is said to be orthonormal ⇒ for w = α1v1 + α2v2 + ⋯⋯⋯⋯ + αnvn, (w, vi) = αi.

UNIT-II

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inner Product Spaces • {vi} is said to be orthonormal ⇒ ∀ w ∈ V, if

u = w − (w, v1)v1 − (w, v2)v2 −⋯⋯⋯⋯− (w, vn)vn, then (u, vi) = 0 ∀ i.

UNIT-II

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Gram-Schmidt Orthogonalization Process : A finite-dimensional inner product space has an orthonormal basis. (in fact, the process is of converting any basis {vi} into orthonormal by using formulae :

UNIT-II

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�w1 = v1/|| v1 ||,

UNIT-II

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�w1 = v1/|| v1 ||,

� .

UNIT-II

27

iiiiii

iiiiiii

vwwvwwvwwv

vwwvwwvwwvw

+),(),(),(

+),(),(),(=

112211

112211

--

--

--------

⋯

⋯



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

28



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inner Product Spaces • If V is a finite-dimensional inner product space

then for any subspace W of V, V = W + W⊥.

UNIT-II

28



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If V is a finite-dimensional inner product space over a field F, then for any subspace W of V, (W⊥)⊥ = W.

UNIT-II

28



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Bessel’s Inequality : If {w1, w2, ⋯⋯⋯⋯ , wn} is an orthonormal set of vectors in an inner product space V over F, then ∀ v ∈ V,

UNIT-II

28

∑ ≤n

i

i vwv1=

22),(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Bessel’s Inequality : If {w1, w2, ⋯⋯⋯⋯ , wn} is an orthonormal set of vectors in an inner product space V over F, then ∀ v ∈ V,

• Parallelogram Law :

|| u + v ||2 + || u − v ||2 = 2(|| u ||2 + || v ||2 )

UNIT-II

28

∑ ≤n

i

i vwv1=

22),(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Fields : Extensions

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Fields : Extensions • If K is a field that contains a field F, then K is

called extension of F.

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Extension is superfield.

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• If K is extension of a field F, the degree of K

over F, denoted by [K : F], the dimension of K as a vector space over F.

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥






• If L is extension of K and K is extension of F, then [L : F] = [L : K][K : F]

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥






• If L is extension of K and K is extension of F, then [L : F] = [L : K][K : F]

• If L is extension of K and K is extension of F, then [K : F] | [L : F].

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Fields : Extensions

UNIT-II

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Fields : Extensions • If K is extension of a field F, then an element

a ∈ K is said to be algebraic element over F ⇔ a satisfies a non-zero polynomial with coefficients in F.

UNIT-II

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If K is extension of a field F and a ∈ K, the F(a) is the smallest subfield of K containing F and a.

UNIT-II

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If K is extension of a field F and a ∈ K, the F(a) is the smallest subfield of K containing F and a.

• If K is extension of a field F, then an element a ∈ K is algebraic element over F ⇔ F(a) is a finite extension of F.

UNIT-II

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Fields : Extensions

UNIT-II

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Fields : Extensions • If K is extension of a field F, then for a ∈ K

the minimal polynomial of a is the non-zero polynomial of the smallest degree satisfied by a.

UNIT-II

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If K is extension of a field F, then a ∈ K is said to be algebraic of degree n over F ⇔ a satisfies a non-zero polynomial of degee n over F and no smaller degree polynomial.

UNIT-II

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If K is extension of a field F, then a ∈ K is said to be algebraic of degree n over F ⇔ a satisfies a non-zero polynomial of degee n over F and no smaller degree polynomial.

• a ∈ K is algebraic of degree n over F ⇒ [F(a) : F] = n.

UNIT-II

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Fields : Extensions

UNIT-II

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Fields : Extensions • If a, b ∈ K are algebraic over F ⇒ a ± b, ab

and a/b (provided b ≠ 0) are all algebraic over F. So, the set of elements of K which are algebraic over F form a subfield of K.

UNIT-II

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If a, b ∈ K are algebraic of degrees m and n, respectively, over F ⇒ a ± b, ab and a/b (provided b ≠ 0) are all algebraic of degree at most mn, over F.

UNIT-II

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If a, b ∈ K are algebraic of degrees m and n, respectively, over F ⇒ a ± b, ab and a/b (provided b ≠ 0) are all algebraic of degree at most mn, over F.

• If K is extension of a field F, such that every element of K is algebraic over F, then K is called algebraic extension of F.

UNIT-II

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Fields : Extensions

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Fields : Extensions • If L is algebraic extension of K and if K is

algebraic extension of F, then L is algebraic extension of F.

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• A complex number is said to be algebraic

number ⇔ it is algebraic over the field of rationals.

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• A complex number which is not algebraic is called transcendental number.

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• A complex number which is not algebraic is called transcendental number.

• π is a transcendental number.

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Algebra of Linear Transformations

UNIT-III

34



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Algebra of Linear Transformations • An algebra A over a field F is an associative

ring which also a vector space over F such that ∀ a, b ∈ A, and ∀ α ∈ F, α(ab) = (αa)b = a(αb).

UNIT-III

34



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If V is a vector space over a field F, then Hom(V, V) is an algebra with unit element over F.

UNIT-III

34



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Hom(V, V) = AF(V) = A(V).

UNIT-III

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Hom(V, V) = AF(V) = A(V).

• Members of A(V) are called as linear transformations.

UNIT-III

34



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

35



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Algebra of Linear Transformations • If A is an algebra with unit element over F,

then A is isomorphic to a subalgebra of A(V) for some appropriate vector space V over F.

UNIT-III

35



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If A is an n-dimensional algebra with unit element over F, then every element of A satisfies a non-trivial polynomial of degree at most n.

UNIT-III

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If A is an n-dimensional algebra with unit element over F, then every element of A satisfies a non-trivial polynomial of degree at most n.

• If A is an algebra with unit element over F, then for a ∈ A, the minimal polynomial of a is the non-zero polynomial of smallest degree satisfied by a.

UNIT-III

35



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

36



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Algebra of Linear Transformations • T ∈ A(V) said to be left invertible ⇔ ∃

S ∈ A(V) such that ST = 1.

UNIT-III

36



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• T ∈ A(V) said to be right invertible ⇔ ∃ S ∈ A(V) such that TS = 1.

UNIT-III

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• T ∈ A(V) said to be invertible/regular ⇔ ∃ S ∈ A(V) such that ST = TS = 1.

UNIT-III

36



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• T ∈ A(V) said to be singular ⇔ it is not invertible.

UNIT-III

36



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• T ∈ A(V) said to be singular ⇔ it is not invertible.

• If V is a finite dimensional vector space over F, then T ∈ A(V) is invertible ⇔ the constant term in its minimal polynomial is not zero.

UNIT-III

36



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

37



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Algebra of Linear Transformations • If V is a finite dimensional vector space over F

and T ∈ A(V) is invertible ⇒ T−1 is a polynomial expression in T over F.

UNIT-III

37



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If V is a finite dimensional vector space over F, then T ∈ A(V) is singular ⇔ ∃ 0 ≠ S ∈ A(V) such that TS = TS = 0.

UNIT-III

37



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If V is a finite dimensional vector space over F, then T ∈ A(V) is singular ⇔ ∃ 0 ≠ S ∈ A(V) such that TS = TS = 0.

• If V is a finite dimensional vector space over F and T ∈ A(V) is right invertible ⇒ T is invertible. (but only for finite dimensional vector space)

UNIT-III

37



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

38



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Algebra of Linear Transformations • If V is a finite dimensional vector space over F,

then T ∈ A(V) is singular ⇔ ∃ 0 ≠ v ∈ V such that vT = 0.

UNIT-III

38



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If V is a vector space over F, then range of T is VT = {vT | v ∈ V}.

UNIT-III

38



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• If V is a vector space over F, then range of T is VT is a subspace of V.

UNIT-III

38



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• If V is a vector space over F, then T ∈ A(V) is invertible ⇔ VT = V, i.e., T is onto.

UNIT-III

38



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• If V is a vector space over F, then T ∈ A(V) is invertible ⇔ VT = V, i.e., T is onto.

• If V is a vector space over F, then for T ∈ A(V) rank of T, denoted by r(T) = dim VT

UNIT-III

38



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Algebra of Linear Transformations • Properties of Rank of rank of linear

transformation :

UNIT-III

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


transformation :

�r(ST) ≤ r(T)

UNIT-III

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


transformation :

�r(ST) ≤ r(T)

�r(TS) ≤ r(T)

UNIT-III

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


transformation :

�r(ST) ≤ r(T)

�r(TS) ≤ r(T)

�r(ST) ≤ min{r(S), r(T)}

UNIT-III

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


transformation :

�r(ST) ≤ r(T)

�r(TS) ≤ r(T)


� If S is invertible, then r(ST) = r(TS) = r(T)

UNIT-III

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


transformation :

�r(ST) ≤ r(T)

�r(TS) ≤ r(T)



� If S is invertible, then r(STS−1) = r(T)

UNIT-III

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


transformation :

�r(ST) ≤ r(T)

�r(TS) ≤ r(T)



� If S is invertible, then r(STS−1) = r(T)

• If V is a vector space over F, then T ∈ A(V) is invertible ⇔ image under T of linearly independent set is linearly independent set.

UNIT-III

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Characteristic Roots

UNIT-III

40



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Characteristic Roots • λ ∈ F is said to be characteristic root of

T ∈ A(V) ⇔ λ − T is singular.

UNIT-III

40



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• λ ∈ F is said to be characteristic root of T ∈ A(V) ⇔ ∃ 0 ≠ v ∈ V such that vT = λv.

UNIT-III

40



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• λ ∈ F is characteristic root of T ∈ A(V) ⇒ for any polynomial q(x) with coefficients in F, q(λ) is characteristic root of q(T).

UNIT-III

40



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• Every characteristic root of T is a root of minimal polynomial of T.

UNIT-III

40



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• Every characteristic root of T is a root of minimal polynomial of T.

• If S ∈ A(V) is regular, the T ∈ A(V) and STS−1 have the same minimal polynomial.

UNIT-III

40



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Characteristic Roots

UNIT-III

41



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Characteristic Roots • λ ∈ F is characteristic root of T ∈ A(V) ⇒ a

non-zero vector v for which vT = λv is called characteristic vector T belonging to characteristic root λ.

UNIT-III

41



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• λ1, λ2, ⋯⋯⋯⋯ , λn ∈ F are distinct characteristic root of T ∈ A(V) and v1, v2, ⋯⋯⋯⋯ , vn are characteristic vectors of T belonging to λ1, λ2, ⋯⋯⋯⋯ , λn, then v1, v2, ⋯⋯⋯⋯ , vn are linearly independent.

UNIT-III

41



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• λ1, λ2, ⋯⋯⋯⋯ , λn ∈ F are distinct characteristic root of T ∈ A(V) and v1, v2, ⋯⋯⋯⋯ , vn are characteristic vectors of T belonging to λ1, λ2, ⋯⋯⋯⋯ , λn, then v1, v2, ⋯⋯⋯⋯ , vn are linearly independent.

• If dim V = n and some T ∈ A(V) has n distinct characteristic roots, then V has a basis consisting only if characteristic vectors of T.

UNIT-III

41



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Matrices

UNIT-III

42



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Matrices • If V is a n-dimensional vector space over F, v1,

v2, ⋯⋯⋯⋯ , vn is a basis of V, then for a T ∈ A(V), matrix of T in basis is (αij) where

UNIT-III

42

∑αn

i

jiji vTv1=

=



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• For a field F, the set Fn of all n × n matrices over F, is an algebra with unit element over F.

UNIT-III

42

∑αn

i

jiji vTv1=

=



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• For a field F, the set Fn of all n × n matrices over F, is an algebra with unit element over F.

• If V is a n-dimensional vector space over F, then V is isomorphic to Fn.

UNIT-III

42

∑αn

i

jiji vTv1=

=



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Matrices

UNIT-III

43



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Matrices • If V is a n-dimensional vector space over F,

T ∈ A(V) has a matrix m1(T) in a basis v1, v2, ⋯⋯⋯⋯ , vn, and a matrix m2(T) in a basis w1, w2, ⋯⋯⋯⋯ , wn, then there exists a matrix C such that m2(T) = Cm1(T)C−1, in fact, one of the values of C can be chosen to be m1(S) where S is linear transformation mapping v1, v2, ⋯⋯⋯⋯ , vn, to w1, w2, ⋯⋯⋯⋯ , wn.

UNIT-III

43



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Matrices • If V is a n-dimensional vector space over F,

T ∈ A(V) has a matrix m1(T) in a basis v1, v2, ⋯⋯⋯⋯ , vn, and a matrix m2(T) in a basis w1, w2, ⋯⋯⋯⋯ , wn, then there exists a matrix C such that m2(T) = Cm1(T)C−1, in fact, one of the values of C can be chosen to be m1(S) where S is linear transformation mapping v1, v2, ⋯⋯⋯⋯ , vn, to w1, w2, ⋯⋯⋯⋯ , wn.

• Using this result, we can obtain different matrices of same linear transformation in different bases of the vector space.

UNIT-III

43



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Linear Algebra LINEAR ALGEBRA

44

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LINEAR ALGEBRA - mathsbscsrtmun.files.wordpress.com€¦ · ॥॥॥ ीह ïर ° °ीीहह ï ïरर °ीह ïर: :: : ॥॥॥॥ Paper No. 10 (B.A.) / 14(B.Sc.)