LinearAlgebra DMTH502

Transcription

1 LiearAlgebra DMTH50

2 LINEAR ALGEBRA

3 Copyright 0 J D Aad All rights reserved Produced & Prited by EXCEL BOOKS PRIVATE LIMITED A-45, Naraia, Phase-I, New Delhi-008 for Lovely Professioal Uiversity Phagwara

4 SYLLABUS Liear Algebra Obectives: This course is desiged for theoretical study of vector spaces, bases ad dimesio, subspaces, liear trasformatios, dual spaces, Elemetary Caoical forms, ratioal ad Jorda forms, ier product spaces, spectral theory ad biliear forms. It should be oted that the successful studet will be able to prove simple theorems i the subect. Sr. No. Descriptio Vector Space over fields, Subspaces, Bases ad Dimesio, Coordiates, Summary of Row-Equivalece, Computatio Cocerig Subspaces Liear Trasformatios, The algebra of liear trasformatios, The traspose of a liear trasformatio, Isomorphism, Represetatio of Trasformatio by matrices 3 Liear Fuctioal, The double dual, Itroductio ad Characteristic Values, Aihilatig Polyomials 4 Ivariat Subspaces, Simultaeous triagulatio, Simultaeous diagoalizatio, Direct-Sum Decompositios 5 Ivariat Direct Sums, The Primary Decompositio Theorem, Cyclic Subspaces ad Aihilators, Cyclic Decompositio ad the ratioal Form 6 The Jorda Form, Computatio of Ivariat Factors, Semi-Simple Operators 7 Ier product, Ier Product Space, Liear Fuctioal ad Adoits, Uitary Operators, Normal Operators 8 Itroductio, Forms o Ier Product Spaces, Positive Forms, More o Forms 9 Spectral Theory, Properties of Normal operators 0 Biliear Forms, Symmetric Biliear Forms, Skew-Symmetric Biliear Forms, Groups Preservig Biliear Forms

5 CONTENTS Uit : Vector Space over Fields Uit : Vector Subspaces 6 Uit 3: Bases ad Dimesios of Vector Spaces 69 Uit 4: Co-ordiates 75 Uit 5: Summary of Row-Equivalece 85 Uit 6: Computatio Cocerig Subspaces 97 Uit 7: Algebra of Liear Trasformatio 06 Uit 8: Isomorphism 8 Uit 9: Represetatio of Trasformatios by Matrices Uit 0: Liear Fuctioals 33 Uit : The Double Dual 44 Uit : Itroductio ad Characteristic Values of Elemetary Caoical Forms 55 Uit 3: Aihilatig Polyomials 67 Uit 4: Ivariat Subspaces 77 Uit 5: Simultaeous Triagulatio ad Simultaeous Diagoalizatio 84 Uit 6: Direct Sum Decompositios of Elemetary Caoical Forms 88 Uit 7: Ivariat Direct Sums 93 Uit 8: The Primary Decompositio Theorem 99 Uit 9: Cyclic Subspaces ad Aihilators 06 Uit 0: Cyclic Decompositio ad the Ratioal Form Uit : The Jorda Form Uit : Computatio of Ivariat Factors 30 Uit 3: Semi-simple Operators 4 Uit 4: Ier Product ad Ier Product Spaces 48 Uit 5: Liear Fuctioal ad Adoits of Ier Product Space 67 Uit 6: Uitary Operators ad Normal Operators 76 Uit 7: Itroductio ad Forms o Ier Product Spaces 90 Uit 8: Positive Forms ad More o Forms 96 Uit 9: Spectral Theory ad Properties of Normal Operators 305 Uit 30: Biliear Forms ad Symmetric Biliear Forms 35 Uit 3: Skew-symmetric Biliear Forms 337 Uit 3: Groups Preservig Biliear Forms 34

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7 Uit : Vector Space over Fields Uit : Vector Space over Fields CONTENTS Obectives Itroductio. Sets. Groups.3 Rigs.4 Fields.5 Vector Spaces.6 Summary.7 Keywords.8 Review Questios.9 Further Readigs Obectives After studyig this uit, you will be able to: Uderstad the cocept of abstract sets Explai the cocept of fuctios Discuss the abstract groups ad their properties State the properties of rigs ad fields Uderstad abstract vector space. This will help you to uderstad sub-spaces, bases ad dimesio i the ext uits Kow that this uit is a prerequisite to uderstad the ext few uits. Itroductio I this uit the idea of set theory is explaied. The uit also deals with fuctios ad mappig. The ideas of rigs ad fields help us to study vector spaces ad their structure. This uit briefly explais the properties of vector spaces which are useful i uderstadig the vector sub-spaces, bases dimesios ad co-ordiates.. Sets The cocept of set is fudametal i all braches of mathematics. A set accordig to the Germa mathematicia George Cator, is a collectio of defiite well-defied obects of perceptio or thought. By a well defied collectio we mea that there exists a rule with the help of which it is possible to tell whether a give obect belogs or does ot belog to the give collectio. The obects i sets may be aythig: umbers, people, aimals etc. The obects costitutig the set are called elemets or members of the set. LOVELY PROFESSIONAL UNIVERSITY

8 Liear Algebra Oe should ote carefully the differece betwee a collectio ad a set. Every collectio is ot a set. For a collectio to be a set, it must be well defied. For example the collectio of ay four atural umbers is ot a set. The members of this collectio are ot well defied. The atural umber 5 may belog or may ot belog to this collectio. But the collectio of the first four atural umbers is a set. Obviously, the members of the collectio are well-defied. They are,, 3 ad 4. A set is usually deoted by a capital letter, such as A, B, C, X, Y, Z etc. ad a elemet of a set by the small letter such as a, b, c, x, y, z etc. A set may be described by actually listig the obects belogig to it. For example, the set A of sigle digit positive itegers is writte as A = {,, 3, 4, 5, 6, 7, 8, 9} Here the elemets are separated by commas ad are eclosed i brackets { }. This is called the tabular form of the set. A set may also be specified by statig properties which its elemets must satisfy. The set is the described as follows: A = {x : P(x)} ad we say that A is the set cosistig of the elemets x such that x satisfies the property P(x). The symbol. is read such that. Thus the set X of all real umbers is simply writte as X = {x : x is real} = {x x is real}. This way of describig a set is called the set builder form of a set. Whe a is a elemet of the set A, we write a A. If a is ot a elemet of A, we write a A. Whe three elemets, a, b ad c, belog to the set A, we usually write a, b, c writig a A, b A ad c A. A, istead of Two sets A ad B are said to be equal iff every elemet of A is a elemet of b ad also every elemet of B is a elemet of A, i.e. whe both the sets cosist of idetical elemets. We write A = B if the sets A ad B are equal ad A B if the sets A ad B are ot equal. If two sets A ad B are such that every elemet of A is also a elemet of B, the A is said to be a subset of B. We write this relatioship by writig A B. If A B, the B is called a superset of A ad we write B A, which is read as B is a super-set of A or B cotais A. If A is ot a subset of B, we write A B, which is read as A is ot a subset of B. Similarly B A is read as B is ot a superset of A. From the defiitio of subset, it is obvious that every set is a subset of itself, i.e., A A. We call B a proper subset of A if, first, B is a subset of A ad secodly, if B is ot equal to A. More briefly, B is a proper subset of A, if B A ad B A. Aother improper subset of A is the set with o elemet i it. Such a set is called the ull set or the empty set, ad is deoted by the symbol. The ull set is a subset of every set, i.e., A. If A is ay set, the the family of all the subsets of A is called the power set of A. The power set of A is deoted by P(A). Obviously ad A are both elemets of P(A). If a fiite set A has elemets, the the power set of A has elemets. LOVELY PROFESSIONAL UNIVERSITY

9 Uit : Vector Space over Fields Example : If A = {a, b, c} the P(A) = {, {a}, {b}, {c}, {a, b} {b, c}, {a, c}, {a, b, c} }. The total umber of these elemets of power set is 8, i.e. 3. The sets A ad B are equal if A is a subset of B ad also B is a subset of A. If U be the uiversal set, the set of those elemets of U which are ot the elemets of A is defied to be the complemet of A. It is deoted by A. Thus Obviously, {A } = A, = U, U =. It is easy to see that if A B, the A B. A = {x : x U ad x A} The differece of two sets A ad B i that order is the set of elemets which belog to A but which do ot belog to B. We deote the differece of A ad B by A ~ B or A B, which is read as A differece B or A mius B. Symbolically A B = {x : x A ad x B}. It is obvious that A A =, ad A = A. Uio ad Itersectio Let A ad B be two sets. The uio of A ad B is the set of all elemets which are i set A or i set B. We deote the uio of A ad B by A B, which is usually read as A uio B. Symbolically, A B = {x : x A or x B} O the other had, the itersectio of A ad B is the set of all elemets which are both i A ad B. We deote the itersectio of A ad B by A B, which is usually read as A itersectio B. Symbolically, A B = {x : x A or x B} or A B = {x : x A, x B}. The uio ad itersectio of sets have the followig simple properties: (i) A B B A ad Commutative laws A B B A (ii) A ( B C) ( A B) C ad Associative laws A ( B C) ( A B) C (iii) A A A ad Idempotet laws A A A (iv) A ( B C) ( A B) ( A C) ad Distributive laws A ( B C) ( A B) ( A C) (v) A ( B C) ( A B) ( A C) ad De Morga's laws A ( B C) ( A B) ( A C) LOVELY PROFESSIONAL UNIVERSITY 3

10 Liear Algebra Two results which iterrelate uio ad itersectio of sets with their complemets are as follows: (i) the complemet of the uio is itersectio of the complemets, i.e., (A B) = A B, ad (ii) the complemet of the itersectio is the uio of the complemets, i.e., (A B) = A B. Suppose A ad B are two sets. The the set (A B) the set A ad B ad is deoted by A B. (B A) is called the symmetric differece of Sice (A B) (B A) = (B A) (A B) Product Set A B = B A. Let A ad B be two sets, a A ad b B. The (a, b) deotes what we may call a ordered pair. The elemet a is called the first coordiate of the ordered pair (a, b) ad the elemet b is called its secod coordiate. If (a, b) ad (c, d) are two ordered pairs the (a, b) = (c, d) iff a = c ad b = d. If A ad B are two sets, the set of all distict ordered pairs whose first coordiate is a elemet of A ad whose secod coordiate is a elemet of B is called the Cartesia product of A ad B (i that order) ad is deoted by A B. Symbolically, A B = {(a, b) : a A ad b B}. I geeral A B B A. If A has elemets ad B has m elemets, the the product set A B has m elemets. If either A or B is a ull set, the A B =. If either A or B is ifiite ad the other is ot empty, the A B is ifiite. We may geeralise the defiitio of the product sets. Let A, A,, A be give sets. The set of ordered -tuples (a, a,, a ) where a A, a A,, a A is called the Cartesia product of A, A,, A ad is deoted by A A A 3 A. Fuctios or Mappigs Let A ad B be two give sets. Suppose there is a correspodece, deoted by f, which associates to each members of A, a uique member of B. The f is called a fuctio or a mappig from A to B. The mappig f from A to B is deoted by f : A B or by A f B. Suppose f is a fuctio from A to B. The set A is called the domai of the fuctio f ad B is called the co-domai of f. The elemet y B which the mappig f associates to a elemet x A is deoted by f (x) ad is called the f-image of x or the value of the fuctio f for x. The elemet x may be referred to as a pre-image of f (x). Each elemet of A has a uique image ad each elemet of B eed ot appear as the image of a elemet i A. There ca be more tha oe elemet of A which have the same image i B. We defie the rage of f to cosist of those elemets of B which appear as the image of at least oe elemet i A. We deote the rage of f : A B by f (A). Thus f (A) = {f (x) : x A}. 4 LOVELY PROFESSIONAL UNIVERSITY

11 Uit : Vector Space over Fields Obviously, f (x) B. If A ad B are ay two o-empty sets, the a fuctio f from A to B is a subset f of A B satisfyig the followig coditio: (i) a A, (a, b) f for some b B; (ii) (a, b) f ad (a, b ) f b = b The first coditio esures that each elemet i A will have image. The secod coditio guaratees that the image is uique. If the domai ad co-domai of a fuctio f are both the same set say f : A called a Operator or Trasformatio of A. A, the f is ofte Two fuctios f ad g of A B are said to be equal iff f (x) = g (x) x A ad we write f = g. For two uequal mappigs from A to B, there must exist at least oe elemet x A such that f (x) g (x). Types of Fuctios If the fuctio f : A B is such that there is at least oe elemet i B which is ot the f-image of ay elemet i A, the we say that f is a fuctio of A ito B. I this case the rage of f is a proper subset of the co-domai of f. If the fuctio f : A B is such that each elemet i B is the f-image of at least oe elemet i A, the we say that f is a fuctio of A oto B. I this case the rage of f is equal to the co-domai of f, i.e., f (A) = B. Oto mappig is also sometimes kow as surectio. A fuctio f : A B is said to be oe-oe or oe-to-oe if differet elemets i A have differet f-images i B, i.e., if f (x) = f (x ) x = x (x ad x A). Oe-to-oe mappig is also sometimes kow as iectio. A mappig f : A B is said to be may-oe if two (or more tha two) distict elemets i A have the same f-image i B. If f : A B is oe-oe ad oto B, the f is called a oe-to-oe correspodece betwee A ad B. Oe-oe oto mappig is also sometimes kow as biectio. Two sets A ad B are said to be have the same umber of elemets iff a oe-to-oe correspodece of A oto B exists. Such sets are said to be cardially equivalet ad we write A ~ B. Let A be ay set. Let the mappig f : A A be defied by the formula f (x) = x, x A, i.e. each elemet of A be mapped o itself. The f is called the idetity mappig o A. We shall deote this fuctio by I A. Iverse Mappig Let f be a fuctio from A to B ad let b B. The the iverse image of the elemet b uder f deoted by f (b) cosists of those elemets i A which have b as their f-image. Let f : A B be a oe-oe oto mappig. The the mappig f : B A, which associates to each elemet b B, the elemet a A, such that f (a) = b is called the iverse mappig of the mappig f : A B. It must be oted that the iverse mappig of f : A it is easy to see that the iverse mappig f : B B is defied oly whe f is oe-oe oto, ad A is also oe-oe ad oto. LOVELY PROFESSIONAL UNIVERSITY 5

12 Liear Algebra Product or Composite of Mappigs Let f : X Y ad g : Y Z. The the composite of the mappigs f ad g deoted by (g o f), is a mappig from X to Z give by (g o f) : X Z such that (g o f) (x) = g [f (x)], x X. If f : X X ad g : X X the we ca fid both the composite mappigs g o f ad f o g, but i geeral f o g g o f. The composite mappig possesses the followig properties: (i) The composite mappig g o f is oe-oe oto if the mappigs f ad g are such. (ii) If f : X Y is a oe-oe oto mappig, the f o f = I y ad f o f = I x. (iii) If f : X Y ad g : Y Z are two oe-oe oto mappigs, ad f : Y X ad g : Z Y are their iverses, the the iverse of the mappig g o f : X Z is the mappig f o g : Z X. (iv) If f : X Y, g : Y Z, h : Z U be ay mappigs, the h o (g o f) ad (h o g) o f are equal mappigs of X ito U, i.e. the composite mappig is associative. Relatio If a ad b be two elemets of a set A, a relatio R betwee them, is symbolically writte as arb, which meas a i R related to b. For example, if R is the relatio >, the statemet a R b meas a is greater tha b. A relatio R is said to be well defied o the set A if for each ordered pair (a, b), where a, b A, the statemet a R b is either true or false. A relatio i a set A is a subset of the product set A A. Iverse Relatio Let R be a relatio from A to B. The iverse relatio of R deoted by R, is a relatio from B to A defied by R y x y B x A x y A B {(, ) :,,(, ) } Clearly, if R is a relatio from A to B, the the domai of R is idetical with the rage of R ad the rage of R is idetical with the domai of R. Differece betwee Relatios ad Fuctios Suppose A ad B are two sets. Let f be a fuctio from A to B. The by the defiitio of fuctio f is a subset of A B i which each a A appears i oe ad oly oe ordered pair belogig to f. I other words f is a subset of A B satisfyig the followig two coditios: (i) for each a A, (a, b) f for some b B, (ii) if (a, b) f ad (a, b ) f, the b = b. O the other had every subset of A B is a relatio from A to B. Thus every fuctio is a relatio but every relatio is ot a fuctio. If R is a relatio from A to B, the domai of R may be a subset of A. But if f is a fuctio from A to B, the domai of f is equal to A. I a relatio from A to B a elemet of A may be related to more tha oe elemet i B. Also there may be some elemets of A which may ot be related to ay elemet i B. But i a fuctio from A to B each elemet of A must be associated to oe ad oly oe elemet of B. 6 LOVELY PROFESSIONAL UNIVERSITY

13 Uit : Vector Space over Fields Equivalece Relatio The relatio R defied o a set A is to be reflexive if ara holds for every a belogig to A, i.e., (a, a) R, for every a A. The relatio R is said to be symmetric if a R b b R a for every ordered pair (a, b) R, i.e., (a, b) R (b, a) R. The relatio R is said to be trasitive if for every a, b, c belogig to A i.e., (a R b, b R c) a R c [(a, b) R, (b, c) R] (a, c) R. A relatio R defied o a set is called a equivalece relatio if it is reflexive, symmetric ad trasitive. Natural Numbers The properties of atural umbers were developed i a logical maer for the first time by the Italia mathematicia G. Peao, by startig from a miimum umber of simple postulates. These simple properties, kow as the Peao s Postulates (Axioms), may be stated as follows: Let there exist a o-empty set N such that. Postulate I: N, that is, is a atural umber. Postulate II: For each N there exists a uique umber + N, called the successor of. Postulate III: For each N, we have +, i.e., is ot the successor of ay atural umber. Postulate IV: If m, N, ad m + = + the m =, i.e. each atural umber, if it is a successor, is the successor of a uique atural umber. Postulate V: If K is ay subset of N havig the properties (i) K ad (ii) m K m + K, the K = N. The postulate V is kow as the Postulate of iductio or the Axiom of iductio. The Priciple of mathematical iductio is ust based o this axiom. Additio Compositio I the set of atural umbers N, we defie additio, which shall be deoted by the symbol + as follows: (i) m + = m + m, N (ii) m + * = (m + )* m, N. The distictive properties of the additio operatio i N are the closure, associative, commutative ad cacellatio laws, i.e., if m,, p N, the (A ) m + N (closure law) (A ) (m + ) + p = m + ( + p) (associative law) LOVELY PROFESSIONAL UNIVERSITY 7

14 Liear Algebra (A 3 ) m + = + m, (commutative law) (A 4 ) m + p = + p m = (cacellatio law) All these properties ca be established from the foregoig postulates ad defiitios oly. Multiplicatio Compositio I the set of atural umbers N, we defie multiplicatio which shall be deoted by the symbol X as follows: (i) m = m m N (ii) m + = m + m, m, N. Sometimes we ofte fid it coveiet to represet m by m. or simply by m. The followig properties, which ca be established from Peao s postulates, hold for multiplicatio. (M ) m, N, or m N, (Closure law) (M ) (m. ). p = m. (. p) or (m ) p = m ( p), (associative law) (M 3 ) m. =. m, or m = m (Commutative law) (M 4 ) m. p =. p m =, or m p = p m =. (Cacellatio law) Distributive Law The distributive property of multiplicatio over additio is expressed i the followig two forms: If m,, p N, we have (i) (ii) m. ( + p) = m. +. p [Left distributive law] (m + ). p = m. p +. p [Right distributive law] The right distributive law ca also be iferred from the left distributive law, sice multiplicatio is commutative. Order Property We say that a atural umber m is greater tha aother umber (m > ), if there exists a umber u N, such that m = + u. The umber m is said to be less tha the umber (m < ), if there exists a umber v = m + v. This order relatio possesses the followig property. N, such that For ay two atural umbers m ad, there exists oe ad oly oe of the followig three possibilities: (i) m = (ii) m >, (iii) m <. This is kow as the Trichotomy law of atural umbers. It is evidet that ay set of atural umbers has a smallest umber, i.e., if A is a o-empty subset of N, there is a umber m A, such that m for every A. 8 LOVELY PROFESSIONAL UNIVERSITY

15 Uit : Vector Space over Fields This is kow as the well orderig property of atural umbers. The relatios betwee order ad additio, ad order ad multiplicatio are give by the followig results: (i) m > m + p > + p, (ii) m > m p > p, for all m,, p N. The operatio of subtractig a umber from aother umber m is possible oly whe m >, i.e., the subtractio operatio is ot defied for ay two atural umbers. It is thus ot a biary compositio i N. Similarly the operatio of dividig oe umber is also ot always possible, i.e., the divisio operatio is also ot a biary compositio i N. Itegers The set of itegers is costructed from the set of atural umbers by defiig a relatio, deoted by ~ (read as wave), i N N as follows: ( a, b) ~ ( c, d) if a d b c, a, b, c, d N. Sice this relatio is a equivalece relatio it decomposes the set N N ito disoit equivalece classes. We defie the set of all these equivalece classes as the set of itegers ad deote it by Z. The equivalece class of the pair (a, b) may be deoted by (a, b) or (a, b)* The additio ad multiplicatio operatios i Z are ow defied as follows: (a, b)* + (c, d)* = (a + c, b + d)* ad (a, b)*. (c, d)* = (ac + bd, ad + bc)*. The associative ad commutative laws of additio ad multiplicatio hold as for atural umbers. The cacellatio law of additio holds i geeral, but the cacellatio law of multiplicatio holds with some restrictios. The distributive law of multiplicatio over additio is also valid. The equivalece class (, )* is defied as the iteger zero, ad is writte as 0. Thus 0 = (, )* = (a, a)* = (b, b)*, a, b N. This umber 0 possesses the properties, that for ay iteger x, (i) x + 0 = x ad (ii) x. 0 = 0. If x = (, )* is a iteger other tha zero, we have, i.e., either > or <. We say that the iteger (, )* is positive if > ad egative if <. Whe >,, N, there exists a atural umber u such that = u +. Therefore a positive iteger x is give by x (, )*,, ( u, )* ( u, ) *. It is possible to idetify the positive iteger (u +, )* with the atural umber u, ad write it as + u. Thus the set of positive itegers may be writte as Z N = {+, +, +3, } LOVELY PROFESSIONAL UNIVERSITY 9

16 Liear Algebra Similarly, a egative iteger ca be idetified with the umber u, ad the set of egative itegers writte as Z N = {,, 3, } We defie the egative of a iteger x as the iteger y, such that x + y = 0. It is easy to see that every iteger has its egative. For, let x = (a, b)*. The if y = (b, a)*, we have x + y = (a, b)* + (b, a)* = (a + b, b + a)* = (a + b, a + b)* = 0 The egative of the iteger x, also called the additive iverse of x, is deoted by x. We therefore have, for ay iteger x, x + ( x) = 0 ad x = (a, b)* x = (b, a)*. We defie subtractio of a iteger if from a iteger x as x + ( y), writte as x y. Thus if x = (a, b)* ad y = (c, d)*, we have Order Relatio i Itegers x y = x + ( y) = (a, b) + (d, c)* = (a + d, b + c)* If x, y be the two itegers, we defie x = y if x y is zero, x > y if x y is positive ad x < y if x y is egative. The Trichotomy Law for itegers holds as for atural umbers. Further, x > y x + z > y + z, ad x > y, z > 0 x z > yz, x, y, z Z. The cacellatio law for multiplicatio states that xz = yz, z 0 x = y. The additio ad multiplicatio operatios o Z satisfy the laws of atural umbers with the oly modificatio i cacellatio law of multiplicatio which requires p 0. Further, the additio operatio satisfies the followig two properties i Z. (i) There exists the additive idetify 0 i the set, i.e., 0 Z such that a + 0 = 0 + a = a, for ay a Z. (ii) There exists the additive iverse of every elemet i Z, i.e., a Z a Z such that a + ( a) = ( a) + a = 0. Divisio A o-zero a is said to be a divisor (factor) of a iteger b if there exists a iteger c, such that b = ac. Whe a is divisor of b, we write a b. Also we say that b is a itegral multiple of a. It is obvious that divisio is ot everywhere defied i Z. The relatio of divisibility i the set of itegers Z is reflexive, sice a a, a Z. It is also trasitive, sice a b ad b c a c. But it is ot symmetric. 0 LOVELY PROFESSIONAL UNIVERSITY

17 Uit : Vector Space over Fields The absolute value a of a iteger a is defied by Thus, except whe a = 0, a Z N. a = a whe a 0 = a whe a < 0 A o-zero iteger p is called a prime if it is either or ad if its oly divisors are,, p, p. Whe a = bc with b > ad c >, we call a composite. Thus every iteger a 0, is either a prime or composite. The operatio of divisio of oe iteger by aother is carried out i accordace with the divisio algorithm, which ca be stated as follows. Give two positive itegers a, b there exists uiquely two o-egative itegers q, r such that a = bq + r, 0 r < b The umber q is called the quotiet, ad r the remaider obtaied o dividig a by b. Two other forms of the theorem, which are successive geeralisatios, are as follows: (i) Give two itegers a, b with b > 0, there exist uique itegers q, r, such that a = bq + r, 0 r < b (ii) Give two itegers a, b with b 0, there exist uique itegers q, r, such that a = bq + r, 0 r < b. Greatest Commo Divisor A greatest commo divisor (GCD) of two itegers a ad b is a positive iteger d such that (i) (ii) d a ad d b, ad if for a iteger c, c a ad c b, the c d. We shall use the otatio (a, b) for the greatest commo divisor of two itegers a ad b. The greatest commo divisor is sometimes also called highest commo factor (HCF). Every pair of itegers a ad b, ot both zero, has a uique greatest commo divisor (a, b) which ca be expressed i the form (a, b) = ma + b for some itegers m ad. Ratioal Numbers Let (a, b) Z Z 0, where Z 0 is the set of o-zero itegers. The the equivalece class ( a, b) {( m, ) : ( m, ) ~ ( a, b); m Z, Z0} is called a ratioal umber. The set of all equivalece classes of Z Z 0 determied by the equivalece relatio ~ defied as above is called the set of ratioal umbers to be deoted by Q. The additio ad multiplicatio operatios i Q are defied as follows: ( a, b) ( c, d) ( ad bc, bd) ad ( a, b) ( c, d) ( ac, bd) LOVELY PROFESSIONAL UNIVERSITY

18 Liear Algebra The associative ad commutative laws of additio ad multiplicatio hold as for itegers, ad so also the distributive law of multiplicatio over additio. The cacellatio laws hold for additio ad multiplicatio, except as for itegers. The additive idetity is the umber (0,). For ( a, b) (0,) ( a. b.) ( a, b) The multiplicative idetity is the umber (,). For, ( a. b) (,) ( a., b.) ( a, b) The additive iverse of ( a, b) is ( a, b ). For, ( a, b) ( a, b) ( a. b ba, b ) (0, b ) (0,) The multiplicative iverse of ( a, b) is ( b, a ) if a 0. For, ( a, b) ( b, a) ( ab, ba) (,) The additive idetity (0,), is defied as the ratioal umber zero ad is writte as 0. The o-zero ratioal umber ( a, b ) which is such that a accordig as a a b is positive or egative. 0, is said to be positive or egative The egative of a ratioal umber z is its additive iverse; it is writte as x. Thus if x = ( a, b) the x = ( a, b ). We defie subtractio of a ratioal umber y from a ratioal umber x as x + ( y), writte x y. Thus, if x = ( a, b ) ad y = ( c, d ), we have x y x ( y) ( a, b) ( c, d) ( ad bc, bd) The reciprocal of a o-zero ratioal umber x is its multiplicative iverse, ad is writte as /x. Thus if x = ( a, b ), the /x = ( b, a ), a 0, b 0. The divisio of a ratioal umber x by a o-zero ratioal umber y, writte as x defied as x. (/y). Thus if x = ( a, b ), the y or x y, is y ( c, d), c 0, we have x y ( a, b) ( d, c) ( ad bc), b 0, c 0. It ca be show that subtractio is a biary compositio i Q, ad divisio is also a biary compositio, except for divisio by zero. Order Relatio Let x, y be two ratioal umbers. We say that x is greater tha, less tha or equal to y, if x y is positive, egative or zero, ad we use the usual sigs to deote these relatios. LOVELY PROFESSIONAL UNIVERSITY

19 Uit : Vector Space over Fields if x = ( a, b ), y = ( c, d ), we have x > y. if x y = ( a, b) ( c, d) ( ad bc, bd ) > 0, whece we fid (ad bc) bd > 0, i.e., ad > bc, b > 0, d > 0. Similarly, x < y if ad < bc, b > 0, d > 0. ad x = y if ad = bc. The Trichotomy Law holds for ratioal umbers, as usual, i.e., give two ratioal umbers x, y either x > y or x = y, or x < y. Also the order relatio is compatible with additio ad multiplicatio. For, x > y x + z > y + z ad x > y, z > 0 xz > yz, x, y, z Q. Represetatio of Ratioal Numbers A ratioal umber of the form ( a,) ca be idetified with the iteger a Z, ad writte simply as a. Further, sice ( a,) ( b,) ( a,) (, b) ( a,, b) ( a, b) we obtai a method of represetig the ratioal umber (a, b) by meas of two itegers. We have ( a, b ) = ( a,) ( b,) = a b or a b, b 0. With this otatio, the sum ad product of two ratioal umbers assume the usual meaig attached to them, viz., a c ad bc b d bd a c ac ad, b 0, d 0 b d bd a c Also ad bc, b 0, d 0. b d The system of ratioal umbers Q provides a extesio of the system of itegral Z, such that (i) Q Z, (ii) additio ad multiplicatio of two itegers i Q have the same meaigs as they have i Z ad (iii) the subtractio ad divisio operatios are defied for ay two umbers i Q, except for divisio by zero. I additio to the properties described above the system of ratioal umbers possesses certai distictive characteristics which distiguish it from the system of itegers or atural umbers. Oe of these is the property of deseess (the desity property), which is described by sayig that betwee ay two distict ratioal umbers there lies aother ratioal umber. LOVELY PROFESSIONAL UNIVERSITY 3

20 Liear Algebra Sice there lies a ratioal umber betwee ay two ratioal umbers, it is clear that there lie a ifiite umber of ratioal umbers betwee two give ratioals. This property of ratioal umbers make them dese every where. Evidetly itegral umbers or the atural umbers are ot dese i this sese. Real Numbers We kow that the equatio x = has o solutio i Q. Therefore if we have a square of uit legth, the there exists o ratioal umber which will give us a measure of the legth of its diagoal. Thus we feel that our system of ratioal umbers is iadequate ad we wat to exted it. The extesio of ratioal umbers ito real umbers is doe by special methods two of which are due to Richard Dedekid ad George Cator. We shall ot describe these methods here. We ca simply say here that a real umber is oe which ca be expressed i terms of decimals whether the decimals termiate at some state or we have a system of ifiite decimals, repeatig or o-repeatig. We kow that every repeatig ifiite decimals is a ratioal umber, also every termiatig decimal is a ratioal umber. Irratioal Number A real umber which caot be put i the form p/q where p ad q are itegers is called a irratioal umber. The set R of real umbers is the uio of the set of ratioal umbers ad the set of irratioal umbers. If a, b, c are real umbers, the (i) a + b = b + a, ab = ba (commutative of additio ad multiplicatio) (ii) a ( b c) ( a b) c, a ( bc) ( ab) c Associativity of additio ad multiplicatio (iii) a + 0 = 0 + a = a, i.e., the real umber 0 is the additive idetity. (iv) a. =.a = a, i.e., the real umber is the multiplicative idetity. (v) For each a R, these correspods a R such that a + ( a) = (a) + a = 0 Thus every real umber has a additive iverse. (vi) Each o-zero real umber has multiplicative iverse. (vii) Multiplicatio compositio distributes additio, i.e., a (b + c) = ab + ac (viii) The cacellatio law ivariably holds good for additio. For multiplicatio, if a ab = ac b = c (ix) The order relatios satisfy the trichotomy law. 0, the Complex Numbers A ordered pair (a, b) of real umbers is called a complex umber. The product set R R cosistig of the ordered pairs of real umbers is called the set of complex umbers. We shall deote the set of complex umbers by C. Thus C = {z : z = (a, b), a, b R}. 4 LOVELY PROFESSIONAL UNIVERSITY

21 Uit : Vector Space over Fields Two complex umbers (a, b) ad (c, d) are equal if ad oly if a = c ad b = d. The sum of two complex umbers (a, b) ad (c, d) is defied to be the complex umber (a + c, b + d) ad symbolically, we write (a, b) + (c, d) = (a + c, b + d) The additio of complex umbers is commutative, associative, admits of idetity elemet ad every complex umber possesses additive iverse. If u ad v are two complex umbers, the u v = u + ( v). The cacellatio law for additio i C is (a, b) + (c, d) = (a, b) + (e, f) (c, d) = (e, f) (a, b), (c, d), (e, f) C. The product of the complex umbers (a, b) ad (c, d) is defied to be the complex umber (ac bd, ad + bc) ad symbolically we write (a, b) (c, d) = (ac bd, ad + bc). The multiplicatio of complex umbers is commutative, associative admits of idetity elemet ad every o-zero complex umber possesses multiplicative iverse. Cacellatio law for multiplicatio i C is I C multiplicatio distributes additio. [(a, b) (c, d) = (a, b) (e, f) ad (a, b) (0, 0)] (c, d) = (e, f) A complex umber (a, b) is said to be divided by a complex umber (c, d) if there exists a complex umber (x, y) such that (x, y) (c, d) = (a, b). The divisio, except by (0, 0), is always possible i the set of complex umbers. Usual Represetatio of Complex Numbers Let (a, b) be ay complex umber. We have (a, b) = (a, 0) + (0, b) = (a, 0) + (0, ) (b, 0) Also, we have (0, ) (0, ) = (, 0) =. If we deote the complex umber (0, ) by i, we have i =. Also we have (a, b) = a + ib, which is the usual otatio for a complex umber. I the otatio Z = a + ib for a complex umber, a is called the real part ad b is called the imagiary parts. A complex umber is said to be purely real if its imagiary part is zero, ad purely imagiary if its part is zero but its imagiary part is ot zero. For each complex umber z = (a, b), we defie the complex umber z = (a, b) to be the cougate of z. I our usual otatio, if z = a + ib the z = a ib If z = (a, b) be ay complex umber, the the o-egative real umber modulus of the complex umber z ad is deoted by z. ( a b ) is called the LOVELY PROFESSIONAL UNIVERSITY 5

22 Liear Algebra. Groups The theory of groups, a importat part i preset day mathematics, started early i ieteeth cetury i coectio with the solutios of algebraic equatios. Origially a group was the set of all permutatios of the roots of a algebraic equatio which has the property that combiatio of ay two of these permutatios agai belogs to the set. Later the idea was geeralized to the cocept of a abstract group. A abstract group is essetially the study of a set with a operatio defied o it. Group theory has may useful applicatios both withi ad outside mathematics. Group arise i a umber of apparetly ucoected subects. I fact they appear i crystallography ad quatum mechaics, i geometry ad topology, i aalysis ad algebra ad eve i biology. Before we start talkig of a group it will be fruitful to discuss the biary operatio o a set because these are sets o whose elemets algebraic operatios ca be made. We ca obtai a third elemet of the set by combiig two elemets of a set. It is ot true always. That is why this cocept eeds attetio. Biary Operatio o a Set The cocept of biary operatio o a set is a geeralizatio of the stadard operatios like additio ad multiplicatio o the set of umbers. For istace we kow that the operatio of additio (+) gives for ay two atural umbers m, aother atural umber m +, similarly the multiplicatio operatio gives for the pair m, the umber m. i N agai. These types of operatios are foud to exist i may other sets. Thus we give the followig defiitio. Defiitio A biary operatio to be deoted by o o a o-empty set G is a rule which associates to each pair of elemets a, b i G a uique elemet a o b of G. Alteratively a biary operatio o o G is a mappig from G G to G i.e. o : G G the image of (a, b) of G G uder o, i.e., o (a, b), is deoted by a o b. G where Thus i simple laguage we may say that a biary operatio o a set tells us how to combie ay two elemets of the set to get a uique elemet, agai of the same set. If a operatio o is biary o a set G, we say that G is closed or closure property is satisfied i G, with respect to the operatio o. Examples: (i) Usual additio (+) is biary operatio o N, because if m, N the m + N as we kow that sum of two atural umbers is agai a atural umber. But the usual substractio ( ) is ot biary operatio o N because if m, N the m may ot belogs to N. For example if m = 5 ad = 6 their m = 5 6 = which does ot belog to N. (ii) Usual additio (+) ad usual substractio ( ) both are biary operatios o Z because if m, Z the m + Z ad m Z. (iii) Uio, itersectio ad differece are biary operatios o P(A), the power set of A. (iv) Vector product is a biary operatio o the set of all 3-dimesioal Vectors but the dot product is ot a biary operatio as the dot product is ot a vector but a scalar. 6 LOVELY PROFESSIONAL UNIVERSITY

23 Uit : Vector Space over Fields Types of Biary Operatios Biary operatios have the followig types:. Commutative Operatio: A biary operatio o over a set G is said to be commutative, if for every pair of elemets a, b G, a o b = b o a Thus additio ad multiplicatio are commutative biary operatios for atural umbers whereas subtractio ad divisio are ot commutative because, for a b = b a ad a b = b a caot be true for every pair of atural umbers a ad b. For example ad 5 4 = Associative Operatio: A biary operatio o o a set G is called associative if a o (b o c) = (a o b) o c for all a, b, c G. Evidetly ordiary additio ad multiplicatio are associative biary operatios o the set of atural umbers, itegers, ratioal umbers ad real umbers. However, if we defie a o b = a b, a, b R the (a o b) oc = (a o b) c = (a b) c = a b c ad a o (b o c) = a (b o c) = a (b c) = a b + 4c. Thus the operatio defied as above is ot associative. 3. Distributive Operatio: Let o ad o be two biary operatios defied o a set, G. The the operatio o is said to be left distributive with respect to operatio o if a o (b o c) = (a o b) o (a o c) for all a, b, c G ad is said to be right distributive with respect to o if, (b o c) o a = (b o a) o (c o c) for a, b, c, G. Wheever the operatio o is left as well as right distributive, we simply say that o is distributive with respect to o. Idetity ad Iverse Idetity: A compositio o i a set G is said to admit of a idetity if these exists a elemet e G such that a o e = a = e o a a G. Moreover, the elemet e, if it exists is called a idetity elemet ad the algebraic structure (G, o) is said to have a idetity elemet with respect to o. Examples: (i) If a R, the set of real umbers the 0 (zero) is a additive idetity of R because a + 0 = a = 0 + a a R N the set of atural umbers, has o idetity elemet with respect to additio because 0 N. LOVELY PROFESSIONAL UNIVERSITY 7

24 Liear Algebra (ii) is the multiplicative idetity of N as a. =.a = a a N. Evidetly is idetity of multiplicatio for I (set of itegers), Q (set of ratioal umbers, R (set of real umbers). Iverse: A elemet a exists b G such that G is said to have its iverse with respect to certai operatio o if there e beig the idetity i G with respect to o. a o b = e = b o a. Such a elemet b, usually deoted by a is called the iverse of a. Thus a o a = e = a o a for a G. I the set of itegers the iverse of a iteger a with respect to ordiary additio operatio is a ad i the set of o-zero ratioal umbers, the iverse of a with respect to multiplicatio is /a which belogs to the set. Algebraic Structure A o-empty set G together with at least oe biary operatio defied o it is called a algebraic structure. Thus if G is a o-empty set ad o is a biary operatio o G, the (G, o) is a algebraic structure. (, +), (I, +), (I, ), (R, +,.) are all algebraic structures. Sice additio ad multiplicatio are both biary operatios o the set R of real umbers, (R, +,.) is a algebraic structure equipped with two operatios. Illustrative Examples Example : If the biary operatio o o Q the set of ratioal umbers is defied by a o b = a + b a b, for every a, b Q show that Q is commutative ad associative. Solutio: (i) o is commutative i Q because if a, b Q, the a o b = a + b a b = b + a b a = b o a. (ii) o is associative i Q because if a, b, c Q the a o (b o c) = a o (b + c b c) = a + (b + c b c) a (b + c b c) = a + b a b + c (a + b a b) c = (a o b) oc. Example 3: Give that S = {A, B, C, D} where A =, B = {a}, ad C = {a, b}. D = {a, b, c} show that S is closed uder the biary operatios (uio of sets) ad (itersectio of sets) o S. 8 LOVELY PROFESSIONAL UNIVERSITY

25 Uit : Vector Space over Fields Solutio: (i) A B = {a} = {a} = B Similarly, A C = C, A D ad A A = A. Also, B B = B, B C = {a} {a, b} = {a, b} = C, B D = {a} {a, b, c} = {a, b, c} = D C C = C, C D = {a, b} {a, b, c} = {a, b, c} = D Hece is a biary operatio o S. (ii) Agai, A A = A, A B = {a} = = A A C = A, A D = A ad B B = B, B C = {a} {a, b} = {a} = B B D = {a} {a, b, c} = {a} = B C C = C, C D = {a, b} {a, b, c} = {a, b} = C. Hece is a biary operatio o S. Self Assessmet. Show that multiplicatio is a biary operatio o the set A = {, } but ot o B = {, 3}.. If A = {, } ad B = {, }, the show that multiplicatio is a biary operatio o A but ot o B. 3. If S = {A, B, C, D} where A =, B = {a, b}, C = {a, c}, D = {a, b, c} show that is a biary operatio o S but is ot. Group Defiitio: A algebraic structure (G, o) where G is a o-empty set with a biary operatio o defied o it is said to be a group, if the biary operatio satisfies the followig axioms (called group axioms). (G ) Closure Axiom: G is closed uder the operatio o, i.e., a o b G, for all a, b G. (G ) Associative Axiom: The biary operatio o is associative, i.e., (a o b) o c = a o (b o c) a, b, G. (G 3 ) Idetity Axiom: There exists a elemet e G such that e o a = a o e = a a G. The elemet e is called the idetity of o i G. (G 4 ) Iverse Axiom: Each elemet of G possesses iverse, i.e., for each elemet a G, there exists a elemet b G such that b o a = a o b = e. The elemet b is the called the iverse of a with respect to o ad we write b = a. Thus a is a elemet of G such that a o a = a o a = e. LOVELY PROFESSIONAL UNIVERSITY 9

26 Liear Algebra Abelia Group of Commutative Group A group (G, o) is said to be abelia or commutative if the compositio o is commutative, i.e., if a o b = b o a a, b G A group which is ot abelia is called o-abelia. Examples: (i) (ii) The structures (N, +) ad (N, ) are ot groups i.e., the set of atural umbers cosidered with the additio compositio or the multiplicatio compositio, does ot form a group. For, the postulate (G 3 ) ad (G 4 ) i the former case, ad (G 4 ) i the latter case, are ot satisfied. The structure (Z, +) is a group, i.e., the set of itegers with the additio compositio is a group. This is so because additio i umbers is associative, the additive idetity O belogs to Z, ad the iverse of every elemet a, viz., a belogs to Z. This is kow as additive group of itegers. The structure (Z, ), i.e., the set of itegers with the multiplicatio compositio does ot form a group, as the axiom (G 4 ) is ot satisfied. (iii) The structures (Q, +), (R, +), (C, +) are all groups i.e., the sets of ratioal umbers, real umbers, complex umbers, each with the additive compositio, form a group. But the same sets with the multiplicatio compositio do ot form a group, for the multiplicative iverse of the umber zero does ot exist i ay of them. (iv) The structure (Q 0, x) is a group, where Q 0 is the set of o-zero ratioal umbers. This is so because the operatio is associative, the multiplicative idetity belogs to Q 0, ad the multiplicative iverse of every elemet a i the set is /a, which also belogs to Q 0. This is kow as the multiplicative group of o-zero ratioals. Obviously (R 0, X) ad (C 0, X) are groups, where R 0 ad C 0 are respectively the sets of ozero real umbers ad o-zero complex umbers. (v) The structure (Q +, ) is a group, where Q + is the set of positive ratioal umbers. It ca easily be see that all the postulates of a group are satisfied. Similarly, the structure (R +, ) is a group, where R + is the set of positive real umbers. (vi) The groups i (ii), (iii), (iv) ad (v) above are all abelia groups, sice additio ad multiplicatio are both commutative operatios i umbers. Fiite ad Ifiite Groups If a group cotais a fiite umber of distict elemets, it is called fiite group otherwise a ifiite group. I other words, a group (G, 0) is said to be fiite or ifiite accordig as the uderlyig set G is fiite or ifiite. Order of a Group The umber of elemets i a fiite group is called the order of the group. A ifiite group is said to be of ifiite order. 0 LOVELY PROFESSIONAL UNIVERSITY

27 Uit : Vector Space over Fields Note: It should be oted that the smallest group for a give compositio is the set {e} cosistig of the idetity elemet e aloe. Illustrative Examples Example 4: Show that the set of all itegers, 4, 3,,, 0,,, 3, 4, is a ifiite abelia group with respect to the operatio of additio of itegers. Solutio: Let us test all the group axioms for abelia group. (G ) Closure Axiom: We kow that the sum of ay two itegers is also a iteger, i.e., for all a, b I, a + b I. Thus I is closed with respect to additio. (G ) Associativity: Sice the additio of itegers is associative, the associative axiom is satisfied, i.e., for a, b, c I. a + (b + c) = (a + b) + c (G 3 ) Existece of Idetity: We kow that O is the additive idetity ad O I, i.e., Hece additive idetity exists. O + a = a = a + O a I (G 4 ) Existece of Iverse: If a I, the a I. Also, ( a) + a = O = a + ( a) Thus every iteger possesses additive iverse. Therefore I is a group with respect to additio. Sice additio of itegers is a commutative operatio, therefore a + b = b + a a, b I. Hece (I, +) is a abelia group. Also, I cotais a ifiite umber of elemets. Therefore (I, +) is a abelia group of ifiite order. Example 5: Show that the set of all eve itegers (icludig zero) with additive property is a abelia group. Solutio: The set of all eve itegers (icludig zero) is I = {0, ±, ± 4, ± 6 } Now, we will discuss the group axioms oe by oe: (G ) The sum of two eve itegers is always a eve iteger, therefore closure axiom is satisfied. (G ) The additio is associative for eve itegers, hece associative axiom is satisfied. (G 3 ) O I, which is a additive idetity i I, hece idetity axiom is satisfied. (G 4 ) Iverse of a eve iteger a is the eve iteger a i the set, so axiom of iverse is satisfied. (G 5 ) Commutative law is also satisfied for additio of eve itegers. Hece the set forms a abelia group. Example 6: Show that the set of all o-zero ratioal umbers with respect to biary operatio of multiplicatio is a group. LOVELY PROFESSIONAL UNIVERSITY

28 Liear Algebra Solutio: Let the give set be deoted by Q 0. The by group axioms, we have (G ) We kow that the product of two o-zero ratioal umbers is also a o-zero ratioal umber. Therefore Q 0 is closed with respect to multiplicatio. Hece, closure axiom is satisfied. (G ) We kow for ratioal umbers. Hece, associative axiom is satisfied. ( a b) c a ( b c) for all a, b, c Q (G 3 ) Sice, the multiplicative idetity is a ratioal umber hece idetity axiom is satisfied. (G 4 ) If a Q 0, the obviously, /a Q 0. Also /a. a = = a. /a so that /a is the multiplicative iverse of a. Thus iverse axiom is also satisfied. Hece Q 0 is a group with respect to multiplicatio. 0 group. Example 7: Show that C, the set of all o-zero complex umbers is a multiplicative Solutio: Let C = {z : z = x + i y, x, y R} Hece R is the set of all real umbers are i = ( ). (G ) Closure Axiom: If a + i b C ad c + id c, the by defiitio of multiplicatio of complex umbers ( a i b) {( c i d) ( a c b d) i ( a d b c) C, sice a c b d, a d b c R, for a, b, c, d R. Therefore, C is closed uder multiplicatio. (G ) Associative Axiom: ( a i b) {( c i d) ( e i f )} ( a c e a d f b c f b d e) i ( a c f a d e b c e b d f ) for a, b, c, d R. = {( a i b) ( c i d)} ( e i f ) (G 3 ) Idetity Axiom: e = (= + i 0 ) is the idetity i C. (G 4 ) Iverse Axiom: Let (a + i b) ( 0) C, the a ib (a + ib) = a ib a b a b i a b a b = a = m + i C, Where m = a b, LOVELY PROFESSIONAL UNIVERSITY

29 Uit : Vector Space over Fields b = a b Hece C is a multiplicative group. Self Assessmet R. 4. Show that the set of all odd itegers with additio as operatio is ot a group. 5. Verify that the totality of all positive ratioals form a group uder the compositio defied by a o b = ab/ 6. Show that the set of all umbers cos + i si forms a ifiite abelia group with respect to ordiary multiplicatio; where rus over all ratioal umbers. Compositio (Operatio) Table A biary operatio i a fiite set ca completely be described by meas of a table. This table is kow as compositio table. The compositio table helps us to verify most of the properties satisfied by the biary operatios. This table ca be formed as follows: (i) (ii) Write the elemets of the set (which are fiite i umber) i a row as well as i a colum. Write the elemet associated to the ordered pair (a i, a ) at the itersectio of the row headed by a i ad the colum headed by a. Thus (i th etry o the left). ( th etry o the top) = etry where the i th row ad th colum itersect. For example, the compositio table for the group {0,,, 3, 4} for the operatio of additio is give below: I the above example, the first elemet of the first row i the body of the table, 0 is obtaied by addig the first elemet 0 of head row ad the first elemet 0 of the head colum. Similarly the third elemet of 4 th row (5) is obtaied by addig the third elemet of the head row ad the fourth elemet of the head colum ad so o. A operatio represeted by the compositio table will be biary, if every etry of the compositio table belogs to the give set. It is to be oted that compositio table cotais all possible combiatios of two elemets of the set will respect to the operatio. : (i) It should be oted that the elemets of the set should be writte i the same order both i top border ad left border of the table, while preparig the compositio table. LOVELY PROFESSIONAL UNIVERSITY 3

30 Liear Algebra (ii) Geerally a table which defies a biary operatio. o a set is called multiplicatio table, whe the operatio is + the table is called a additio table. Group Tables The compositio tables are useful i examiig the followig axioms i the maer explaied below:. Closure Property: If all the elemets of the table belog to the set G (say) the G is closed uder the Compositio o (say). If ay of the elemets of the table does ot belog to the set, the set is ot closed.. Existece of Idetity: The elemet (i the vertical colum) to the left of the row idetical to the top row (border row) is called a idetity elemet i the G with respect to operatio o. 3. Existece of Iverse: If we mark the idetity elemets i the table the the elemet at the top of the colum passig through the idetity elemet is the iverse of the elemet i the extreme left of the row passig through the idetity elemet ad vice versa. 4. Commutativity: If the table is such that the etries i every row coicide with the correspodig etries i the correspodig colum i.e., the compositio table is symmetrical about the pricipal or mai diagoal, the compositio is said to have satisfied the commutative axiom otherwise it is ot commutative. The process will be more clear with the help of followig illustrative examples. Illustrative Examples Example 8: Prove that the set of cube roots of uity is a abelia fiite group with respect to multiplicatio. Solutio: The set of cube roots of uity is G = {,, }. Let us form the compositio table as give below: I 3 = 3 = 4 = (G ) Closure Axiom: Sice each elemet obtaied i the table is a uique elemet of the give set G, multiplicatio is a biary operatio. Thus the closure axiom is satisfied. (G ) Associative Axiom: The elemets of G are all complex umbers ad we kow that multiplicatio of complex umber is always associative. Hece associative axiom is also satisfied. (G 3 ) Idetity Axiom: Sice row of the table is idetical with the top border row of elemets of the set, (the elemet to the extreme left of this row) is the idetity elemet i G. (G 4 ) Iverse Axiom: The iverse of,, are, ad respectively. (G 5 ) Commutative Axiom: Multiplicatio is commutative i G because the elemets equidistat with the mai diagoal are equal to each other. The umber of elemets i G is 3. Hece (G,.) is a fiite group of order 3. 4 LOVELY PROFESSIONAL UNIVERSITY