Syllabus S.Y.B.Sc. (C.S.) Mathematics Paper II Linear Algebra

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1 Syllabus S.Y.B.Sc. (C.S.) Mathematics Paper II Liear Algebra Uit : Systems of liear equatios ad matrices (a) Systems of homogeeous ad o-homogeeous liear equatios (i) The solutios of systems of m homogeeous liear equatios i ukows by elimiatio ad their geometric iterpretatio for m,,, 3,, 3 3, 3. (ii) The existece of o-trival solutio of such a system for m. The sum of two solutios ad a scalar multiple of a solutio of such a system is agai a solutio of the system. (b) (i) Matrices over, The matrix represetatio of systems of homogeeous ad o-homogeeous liear equatios. (ii) Additio, scalar multiplicatio ad multiplicatio of matrices, Traspose of a matrix (iii)the types of matrices : zero matrix, idetity matrix, symmetric ad skew symmetric matrices, upper ad lower triagular matrix. (iv) Traspose of product of matrices, Ivertible matrices, Product of ivertible matrices. (c) (i) Elemetary row operatios o matrices, row echelo from of a matrix ad Gaussia elimiatio method. Applicatios of Gauss elimiatio method to solve system of liear equatios. (ii) The matrix uits, Row operatios ad Elemetary matrices, Elemetary matrices are ivertible ad a ivertible matrix is a product of elemetary matrices. Referece for Uit : Chapter II, Sectios,, 3, 4, 5 of Itroductio to Liear Algebra, SERGE LANG, Spriger Verlag ad Chapter, of Liear Algebra A Geometric Approach. S. KUMARESAN, Pretice Hall of Idia Private Limited, New Delhi. Uit : Vector spaces over (a) Defiitio of a vector space over. Examples such as (i) Euclidea space. (ii) The space of sequeces over. (iii) The space of m matrices over. (iv) The space of polyomials with real coefficiets. (v) The space of real valued fuctios o a o-empty set. (b) Subspaces defiitio ad examples icludig

2 (i) (c) (i) Lies i, Lies ad plaes i 3. (ii) The solutios of homogeeous system of liear equatios, hyperplae. (iii) The space of coverget real sequeces. (iv) The spaces of symmetric, skew symmetric, upper triagular, lower triagular, diagoal matrices. (v) The space of polyomials with real coefficiets of degree. (vi) The space of cotiuous real valued fuctios o a, b. (vii) The space of cotiuously differetiable real valued fuctios o a, b. The sum ad itersectio of subspaces, direct sum of a subset of a vector space. (ii) Liear combiatio of vectors, covex sets, liear spa of subset of a vector space. (iii) Liear depedece ad idepedece of a set. (d) (The discussio of cocepts metioed below for fiitely geerated vector spaces oly) Basis of a vector space, basis as a maximal liearly idepedet set ad a miimal set of geerators. Dimesio of a vector space. (e) (i) Row space, Colum space of a m matrix over ad row rak, colum rak of a matrix. (ii) Equivalece of row rak ad colum rak, Computig rak of a matrix by row reductio. Referece for Uit : Chapter III, Sectios,, 3, 4, 5, 6 of Itroductio to Liear Algebra, SERGE LANG, Spriger Verlag ad Chapter of Liear Algebra A Geometric Approach S. KUMARESAN, Pretice Hall of Idia Private Limited, New Delhi. Uit 3 : Ier Product Spaces (a) Dot product i space over., Defiitio of geeral ier product o a vector Examples of ier product icludig the ier product f. g f t g t dt C,, the space of cotiuous real o valued fuctios o,. (b) (i) Norm of a vector i a ier product space.

3 Cauchy-Schwarz iequality, triagle iequality. 3 (ii) Orthogoality of vectors, Pythagorus theorem ad geometric applicatios i. (iii) Orthogoal complemets of a subspace, Orthogoal 3 Complemets i ad. (iv) Orthogoal sets ad orthoormal sets i a ier product space. Orthogoal ad orthoormal bases. Gram-Schmidt orthogoalizatio process, simple examples i 3,. Referece for Uit 3 : Chapter VI, Sectios, of Itroductio to Liear Algebra, SERGE LANG, Spriger Verlag ad Chapter 5 of Liear Algebra A Geometric Approach, S. KUMARESAN, Pretice Hall of Idia Private Limited, New Delhi. Uit 4 : Liear Trasformatios (a) Liear trasformatios defiitio ad properties, examples icludig (i) (b) (i) (c) (i) (d) (i) Natural projectio from (ii) The map L : A m matrix over. (iii) Rotatios ad reflectios i (iv) Orthogoal projectios i (v) Fuctioals. to m. m m defied by A L X AX, where A is a, Strechig ad Shearig i.. The liear trasformatio beig completely determied by its values o basis. The sum ad scalar multiple of liear trasformatios from U to V where U, V are fiite dimesioal vector spaces over is agai a liear trasformatio. (ii) The space LU, V of liear trasformatios from U to V. (iii) The dual space V where V is fiite dimesioal real vector space. Kerel ad image of a liear trasformatio (ii) Rak-Nullity Theorem (iii) The liear isomorphisms, iverse of a liear isomorphism (iv) Composite of liear trasformatios Represetatio of a liear trasformatio from U to V, where U ad V are fiite dimesioal real vector spaces by matrices with

4 4 respect to the give ordered bases of U ad V. The relatio betwee the matrices of liear trasformatio form U to V with respect to differet bases of U ad V. (ii) Matrix of sum of liear trasformatios ad scalar multiple of a liear trasformatio. (iii) Matrices of composite liear trasformatio ad iverse of a liear trasformatio. (e) Equivalece of rak of a m matrix A ad rak of the liear m trasformatio LA: LA X AX. The dimesio of solutio space of the system of liear equatios AX 0 equals rak A. (f) The solutios of o-homogeeous systems of liear equatios represeted by AX B. (i) Existece of a solutio whe rak A = rak A, B. (ii) The geeral solutio of the system is the sum of a particular solutio of the system ad the solutio of the associated homogeeous system. Referece for Uit 4 : Chapter VIII, Sectios, of Itroductio to Liear Algebra, SERGE LANG, Spriger Verlag ad Chapter 4, of Liear Algebra A Geometric Approach, S. KUMARESAN, Pretice Hall of Idia Private Limited, New Delhi. Uit 5 : Determiats (a) Defiitio of determiat as a -liear skew-symmetric fuctio from... E, E,..., E is such that determiat of, where j E deotes the j th colum of the idetity matrix I. Determiat of a matrix as determiat of its colum vectors (or row vectors) (b) (i) Existece ad uiqueess of determiat fuctio via permutatios. (ii) Computatio of determiat of, 3 3 matrices, diagoal matrices. (iii) Basic results o determiats such as t det A det A, det AB det A det B. (iv) Laplace expasio of a determiat, Vadermode determiat, determiat of upper triagular ad lower triagular matrices.

5 (c) (i) Liear depedece ad idepedece of vectors i determiats. 5 usig (ii) The existece ad uiqueess of the system AX B, where A is a det A 0. matrix with (iii) Cofactors ad miors, Adjoit of a matrix A. A. adj A det A. I. A real matrix A is ivertible if ad oly if Basic results such as det A 0; A adj A for a ivertible matrix A. det A (iv) Cramer s rule (d) Determiat as area ad volume Referece for Uit 5 : Chapter VI of Liear Algebra A geometric approach, S. KUMARSEAN, Pretice Hall of Idia Private Limited, 00 ad Chapter VII Itroductio to Liear Algebra, SERGE LANG, Spriger Verlag. Uit 6 : Eigevalues ad eigevectors (a) (i) Eigevalues ad eigevectors of a liear trasformatio T : V V, where V is a fiite dimesioal real vector space. (ii) Eigevalues ad eigevectors of real matrices ad eigespaces. (iii) The liear idepedece of eigevectors correspodig to distict eigevalues of a matrix (liear trasformatio). (b) (i) The characteristic polyomial of a real matrix, characteristic roots. (ii) Similar matrices, characteristic polyomials of similar matrices. (c) The characteristic polyomial of a liear trasformatio T : V where V is a fiite dimesioal real vector space. V, Referece for Uit 6 : Chapter VIII, Sectio, of Itroductio to Liear Algebra, SERGE LANG, Spriger Verlag ad Chapter 7, of Liear Algebra A Geometric Approach, S. KUMARESAN, Pretice-Hall of Idia Private Limited, New Delhi. The proofs of the results metioed i the syllabus to be covered uless idicated otherwise. Recommeded Books :. SERGE LANG : Itroductio to Liear Algebra, Spriger Verlag.. S. KUMARESAN : Liear Algebra A Geometric approach, Pretice Hall of Idia Private Limited.

6 6 Additioal Referece Books :. M. ARTIN : Algebra, Pretice Hall of Idia Private Limited.. K. HOFFMAN ad R. KUNZE : Liear Algebra, Tata McGraw Hill, New Delhi. 3. GILBERT STRANG : Liear Algebra ad its applicatios, Iteratioal Studet Editio. 4. L. SMITH : Liear Algebra, Spriger Verlag. 5. A. RAMACHANDRA RAO ad P. BHIMA SANKARAN : Liear Algebra, Tata McGraw Hill, New Delhi. 6. T. BANCHOFF ad J. WERMER : Liear Algebra through Geometry, Spriger Verlag New York, SHELDON AXLER : Liear Algebra doe right, Spriger Verlag, New York. 8. KLAUS JANICH : Liear Algebra. 9. OTTO BRETCHER : Liear Algebra with Applicatios, Pearso Educatio. 0. GARETH WILLIAMS : Liear Algebra with Applicatios, Narosa Publicatio. Suggested topics for Tutorials / Assigmets : () Solvig homogeeous system of m equatios i ukows by m,,, 3,, 3 3, 3. elimiatio for () Row echelo from, Solvig system AX B by Gauss elimiatio. (3) Subspaces : Determiig whether a give subset of a vector space is a subspace. (4) Liear depedece ad idepedece of subsets of a subsets of a vector space. (5) Fidig bases of vector spaces. (6) Rak of a matrix. (7) Gram-Schmidt method. (8) Orthogoal complemets of subspaces of (9) Liear trasformatios. 3 (lies ad plaes). (0) Determiig kerel ad image of liear trasformatios.

7 7 () Matrices of liear trasformatios. () Solutios of system of liear equatios. (3) Determiats : Computig determiats by Laplace s expasio. (4) Applicatios of determiats : Cramer s rule. (5) Fidig iverses of, 3 3 ivertible matrices usig adjoit. (6) Fidig characteristic polyomial, eigevalues ad eigevectors of ad 3 3 matrices. (7) Fidig characteristic polyomial, eigevalues ad eigevectors of liear trasformatios.

8 8 SYSTEMS OF LINEAR EQUATIONS AND MATRICES Uit structure.0 Objectives. Itroductio. Systems of Liear equatios ad matrices.3 Equivalet Systems.4 Exercise.5 Uit Ed Exercise.0 OBJECTIVES Our aim i this sectio is to study the rectagular arrays of umbers correspodig liear equatio ad some related cocept to develop a elemetary theory of the same. Object of this sectio is to solve the ukows providig differet ad simplest methods for uderstadig of the studets i a very simple maer.. INTRODUCTION I this sectio, we shall be cocered with sets of umbers arraged systematically i the colums ad rows, called rectagular arrays. To take a example cosider the system of liear equatios: a x a x a x 0; 3 3 a x a x a x 0; 3 3 a x a x a x We shall see that while solvig this system of equatios, we i fact, fid ourselves workig oly with the rectagular array of costats : a a a a a a a a a

9 9 The ukows x, x, x3 act merely as marks of positio i the equatios. Also i this sectio we itroduce basic termiology ad discuss a method for solvig system of liear equatios. A lie i the xy plae ca be represeted algebraically by a equatio of the form. a x a x b, A equatio of this kid is called a liear equatio i the variables x ad x. More geerally, we defie a liear equatio i the variables x, x,..., x to be oe that ca be expressed i the form. a x a x... a x b, where a, a,..., a ad b are real umbers.. SYSTEMS OF LINEAR EQUATIONS AND MATRICES : We shall deal with the problems of solvig systems of liear equatios havig ukows. There are two aspects to fidig the solutios of liear equatios. Firstly, the formal maipulative aspect of computatios with matrices ad secodly, the geometric iterpretatio of the system as well as the solutios. We shall deal with both the aspects ad itegrate algebraic cocepts with geometric otios. System of liear equatios : The collectio of liear equatios ax a x... a x b ax ax... ax b amx am x... amx bm (*) where a ij, bi, i m, j. is called a system of m liear equatios i ukow. If b b... b m 0

10 0 i.e. a x a x... a x 0 ax ax... ax 0 am x am x... amx 0 It is called a homogeeous system of m liear equatios i ukows. If atleast oe bi 0, where i m, it is called a ohomogeeous system of m liear equatios i ukows. i.e. If a x a x... a x 3 ax ax... ax 0 am x am x... amx 0 is called a o-homogeeous system of m liear equatios. I short, Simple example x y 3z 0 x 3y 4z 0 x y 5z 0 (homogeeous system) is called a homogeeous systems i three variables x, y ad z. If x y 3z 5 x 3y z 3 x y 5z 0 (o-homogeeous system) is called o-homogeeous system i three variables x, y ad z. ij j i j We may represet (*) i a short form as a x b, i m. Ay -typle x x x,..., which satisfies all the equatios i the system (*) is called a solutio of the system. xi, i. The set of solutios is called the solutio set or sometimes, the geeral solutios of the system. If the system has o solutio, we say the system is icosistet. We give few examples :. x + y = 4 E x + y = 6 E (icosistet)

11 . x y 4 E x 3y 7 E (uique solutio) 3. x y 4 E x 4y 8 E (ifiite may solutios) We try to solve the above systems by elimiatig oe variable. I system (). E E gives O =, thus the system is icosistet. I system (), E E gives y =, substitutig y = i E, we get x =. Thus we have a uique solutio set, for system (). I system (3), the equatio (E ) is obtaied by multiplyig the first equatio by. Therefore (s, t) is a solutio of this system, NOTE : If m, the system AX 0 has a o-trivial solutio. s t 4 s 4t 8 4 t, t is a solutio of (3) for ay t. The solutio set is 4 t, t : t. Thus system (3) has ifiitely may solutios. Geometrically, the first system has two parallel lies which do ot itersect. The secod system has two itersectig lies itersectig i oe poit. The third system has oe lie ad every poit o the lie is a solutio. Thus, the system has ifiitely may solutios. Thus, a system of o homogeeous equatios may have o solutios, a uique solutio or ifiitely may solutios. However, a system of homogeeous equatios i ukows, where all c i = 0 i This is called a trivial solutio. Ay solutio c,..., c where at least oe i 0 solutio. c, i is called a o-trivial We ote that i order to solve the give system of equatios, we elimiate oe variable. The most fudametal techique for fidig the solutios of a system of liear equatios is the techique of elimiatio. We multiply a equatio by a o-zero scalar ad add it to aother equatios to produce a equatio where oe of the ukows may be abset. We assume that the ew system obtaied has same solutios as the origial system.

12 .3 EQUIVALENT SYSTEMS : Cosider the system of liear equatios. aijx j bi ; for i m j A system of equatios obtaied by i) Multiplyig a equatio i the system by a o-zero scalar. ii) Addig scalar multiple of a equatio i the system to aother equatio is called asystem equivalet to the give system. Problem: We ow fid solutio of m homogeeous liear equatios m,,, 3,, 3 3, 3. i ukows whe i) m =, = If 0 ax a y 0, atleast o of a,a 0. a, ad s,t a s at 0 is a solutio of the system. S a / a t, t The solutio set is a t, t t a a i.e.,t a t If a 0, the a 0 ad multiplyig the equatios by get equatio y 0. a we The solutio set is t, 0 t or 0 system has ifiitely may solutios., t t. Thus the Geometrically, the system represets a straight lie through origi ad every poit o the lie is a solutio.

13 3 a xa y0 m = = ii) ax a y 0 E ax ay 0 E a If a λ, λ λ 0, the multiplyig equatio (i) by λ, we a a get that two equatios are same ad the system is ax a y 0, which a we discussed i (i) as a, the a aa a 0. a a If s,t is a solutio the a times E a times E gives aa a a y 0 i.e. a a aa t 0 t 0 a a aa 0 Substitutig i ( E ) ad (E ) we get a xa y0 as 0 as 0 s = 0 a xa y0

14 4 The system has uique solutio, amely trivial solutio (Geometrically, the system represets two lies passig through origi. iii) m =, = 3 a x a y a z 3 0 If a 0 ad (r, s, t) is a solutio of the above system ar a s a3t 0. a a3t r s, a a s,t. The solutio set is a a3 s t, s, t s, t a a a i.e. a,, 0 s 3, 0, t s, t a a Thus, the system has ifiitely may solutios. Geometrically the system represets a plae passig through origi ad ay poit o the plae is a solutio of the system. iv) m =, = 3 ax a y a3 z 0 ( E ) ax ay a3z 0 ( E ) a Let a a 3 λ, the ( E ) ad ( E ) are same equatios. a a a 3 (multiplyig E by λ ) ad we have already discussed this i (iii). a If a a or a3 a or a 3 a a a a3 a a3 We elimiate x by ae a3 E. Similarly, we elimiate y by ae a E, ad z by a3e a3 E. Thus we get x y z aa3 aa 3 aa3 aa3 aa aa If (r, s, t) is a solutio of the system. r s t If aa3 aa λ, for λ 3 aa3 aa 3 aa aa λ = λ a a a a, a a a a, a a a a λ The solutio set is λa a a a At least oe of the x, y, z co-ordiate is o-zero ad we have ifiitely may solutios. Geometrically, the system represets two plaes passig through origi ad the plaes itersect i a lie through origi.

15 5 v) m 3, 3 ax a y a3 z 0 ( E ) ax ay a3z 0 ( E ) a3x a3 y a33z 0 ( E 3 ) If ay equatio i the above system is a liear combiatio of the other two equatios the system is equivalet to a system of oe equatio i three ukows or two equatios i three ukows ad these cases have bee discussed. Otherwise a a a3 a a a3 0 a3 a3 a33 If r, s, t is a solutio of the above system, we cosider equatio (E ) ad (E 3 ) ad get r s t aa33 a3a 3 a3a3 aa33 aa3 aa3 λ, for λ Substitutig i E aλ aa33 a3a3 a λa3a3 aa33 a3 λaa3 a3 a 0 a a a3 λ a a a3 0 ad so λ 0. a3 a3 a33 r, s, t 000,, Geometrically, the three equatios represet three plaes passig through origi ad they itersect i a uique poit, ame the origi. We observe that, i the above systems, the system has ifiitely may solutios of m <. But, for m =, the system has oly trival solutio provided certai coditios were satisfied by the coefficiets a ij. NOTE : The system aijxj 0, i m of m homogeeous liear j equatios i -ukows has a o-trivial solutio if m <. Solve examples : ) Fid atleast oe o-trivial solutio of the system 3x y z 0.

16 6 Solutio : We take y, z ad get 3,, is a o-trivial solutio of the system. x the 3 3 ) Write the geeral solutio of the system ad give geometrical iterpretatio. x 3y 4z 0 3x y z 0 Solutio : The give system is x 3y 4z 0 ( E ) 3x y z 0 ( E ) To elimiate y, by multiplyig E by 3 ad addig to E, we get x7z 0 ( E ) x 3y 4z 0 ( E ) 7 If r, s, t is a solutio of the system, the r7t 0, i.e. r t Substitutig i E, we get r 3s 4t 0 4t i.e. 3s 4t 0 30t 0t 3s ; s 7 0t Thus, the solutio is t,,t,t. 7 0 The set of solutio is,, t t This represets a lie passig through origi ad havig directio 7 0,, i 3. The give system is represetig two plaes i 3 passig through (0, 0, 0). They iterestig lie passig through 0, 0, 0. 3) Show that the oly solutio of the followig system is the trivial solutio. 4x 7y 3z 0 x y 0 y 6z 0 Solutio : The give system is 4x 7y 3z 0 ( E ) x y 0 ( E ) y 6z 0 ( E 3 )

17 7 To elimiate y by performig E E3 we get 4x 7y 3z 0 ( E ) x y 0 ( E ) x 6z 0 ( E 3 ) If r, s, t is a solutio of the system, E gives r + s = 0 i.e. r s. E 3 gives r + 6t = 0 i.e. r t. 6 E gives 4r 7s 3t 0 r i.e. 4r 7r r 0 r s r 0, t 0. Thus the oly solutio of the system is 0, 0, 0. 6 Tutorial / Assigmet : Topic :- Solvig homogeeous system of m liear equatios i m,,,, 3,,,, 3, 3, 3. ukows by elimiatio for Theorem :- Show that the system of liear equatios. aijx j 0, i, m has a o-trivial solutio for j m,,,, 3 ad 3,. Theorem :- Show that the system of liear equatios aijx j 0, i m has uique solutio for m,, ad 33,. j If a a a a a a a 3 a a a 3 a a a (for m ) ad 0 (for m 3) Exercises : Show that the followig systems of liear equatios have ifiitely may solutios. Write the solutio set ad the geometrical iterpretatios of the system as well as solutio. i) x y z 0 ii) x y z 0 3x y 4z 0

18 8 iii) 3x y z 0 x y z 0 Exercises : Show that the followig systems of equatios have oly trivial solutio. i) x 3y 0 x y 0 ii) 4x 5y 0 6x 7y 0 iii) 3x 4y z 0 x y z 0 x 3y 5z 0 We observe that :. Propositio : For a homogeeous system of m liear equatios i ukows, the sum of two solutios ad a scalar multiple of a solutio is agai a solutio of the same system.. Propositio : A ecessary ad sufficiet coditio for the sum of two solutios or ay scalar multiple of a solutio to be the solutio of the same system of liear equatios aijx j bi, i m is that b i 0 j for i m. System of Liear Equatios : Geeral form of the system of liear equatio is ax ax... a x b ax ax... ax b amx amx... amx bm (A) This is called as a system of m equatio i -ukow variables. Here a ij s ad b i s are real umbers ad x, x,..., x are ukows aij ' s are called costats. Examples x 5y 7z w8 7 3y 4w8 3x 5z 9w 0 is a system of 4 variables. The above system (A) ca be writte as

19 a x b, i m 9 ij j i j The system (A) ca be writte i the matrix form as a a a x b a a a x b a a a x b m m m m m Agai we ca write this as AX B; where A is a matrix of order m also it is called as matrix of coefficiet. X is a colum matrix of variables ad B is a colum matrix of costat. Theorem : If c c, c,..., c ad c c, c,..., c are solutios of homogeeous system AX 0. a) b) aijx j 0, i m the j. c c, c,..., c, is also a solutio. c c c c, c c,..., c c is also a solutio. Proof (a) : Sice c c c c we have,,..., is the solutio of the system therefore, aijc j 0, i m () j Cosider L.H.S. = aijxj = aij cj =. aijcj j j j =.0 [from equatio ()] = 0 = R.H.S.. c c, c,..., c is the solutio of the give system. b) Sice c c,c,...,c is the solutio of the give system. We have cijc j 0, i m () j Cosider L.H.S. = aijxj = j a ij c j c j = j aijc j aijc j j j = 0 [from () ad ()] = R.H.S. c c c c, c c,..., c c is the solutio of the system.

20 0 Theorem : Cosider a system of m-equatio i ukows i.e. a system aijx j 0, i m. If m, the the system has otrivial j solutio. Proof : We prove the result by iductio method o m. Let m,. The system of equatio reduce to a x a x... a x 0. Let a p 0, p. Cosider, ax a x... a pxp... a x 0 (I) Let x, x 0, x3 0,..., xp 0 a x p, x p 0,..., x 0 a p Substitutig the above values of x, x,..., x i RHS of (I). We get, L.H.S. of (I) which is equal to zero. a x, x 0,..., xp,.., x 0 is the o-trivial solutio of (I). a p for m the system has o-trivial solutio. result is true for m. Let the result be true for m k. i.e. the result is true for system of k umber of equatio. The to prove that result is true for m = k + ad > k +. Cosider a system of k + equatio. a x a x... a x E a x a x... a x E 0 0 a k a k x... a k x 0 I E, all the coefficiets are ot equal to zero. Ek Let a p 0 for some p, p r. from E ax a x... a px p a pxp a px p... a x 0 Sice a p 0, we get a a a a 3 p xp x x x 3... x p... a p a p a p a p ap a x p... x (II) a p a p

21 Substitutig the values of x p i remaiig equatio E, E 3,..., Ek, we get, the system k equatios. I variables x,x,...,x p,x p,...,x we deote there equatios by 3 k E, E,.., E & E k. This is the system of k equatios therefore, by iductio it has o-trivial solutio. Let x c, x c,...,x p c p, xp c p,...,x c be the otrivial solutio of the above system. Substitutig the values of x,x,...,x p,..,x i (II), we get the values x p. Therefore, we get the values of x,x,...,x which are ot all zeros. Therefore the system of k equatio has o-trivial solutio. Note : i) If, m, the system has o-trivial solutio but the coverse may ot be true. i.e. if the system has o-trivial the it does ot mea m. ii) If c c,c,...,c is the o-trivial solutio of homogeeous system tha., c is also a solutio. If the system has o-trivial solutio the it has ifiitely may solutio. Examples : ) Fid the solutio set of the followig problems. a) x 3y 5z 0 (i) x y z 0 (ii) Solutio : (ii) y x z x 3 x z 5z 0. Substitutig y, i (i), we get x z Agai, y x z z z 3z y 3z Solutio set is z, 3, z x,y,z / x,y,z 3 z, z, z z

22 EXERCISE: i) x 4z 3y 0 3x z y 0 ii) x y 4z w 0 3x y 3z w 0 x y z 0 iii) x y z 3w 0 x 4z 4w 0 x y z w 0 x y z 0 iv) 7x y 5z w 0 x y z 0 x z w 0 y z w 0 Example : x 3y 4z 0 () 3 x y z 0 () Solutio : () y 3 x z Substitutig y i (), we get z 7 x Sice y 3 x z 3x 7 x 0x y 0 x Solutio set is S x, y, z : x, y, z x,0, 7 : x x,0 x,7 x : x Example 3: x y 4z w 0 () 3x y 3z w 0 () x y z 0 (3) Solutio : (3) y x z Substitutig y x z i (), we get w 5x 5 z Substitutig the values of y ad w i (), we get 4 x z S z,,, z Example 4: 7x y 5z w 0 ()

23 3 x y z 0 () x z w 0 (3) y z w 0 (4) () y x z (5) By substitutig o (), we get w 5 x 3 z (6) (3) z x x 0 (5) y 0 (6) w 0 Solutio set is S x, y, z, w x, y, z, w S 0, 0, 0, 0..4 EXERCISE : Solve the followig homogeeous systems by ay method : i) x x 3x3 0 x x 0 x3x3 0 As : x, x, x3 : x, x, x3 i.e. 0, 0, 0 ii) 3x x x3 x4 0 5x x x3 x4 0 As : x s, x t s, x3 4 s, x4 t iii) x y 4z 0 w y 3z 0 w 3x y z 0 w x 3y z 0 As : w t, x t, y t, z 0 iv) x y 3z 0 x y 3z 0 x y 4z 0 As : (Oly the trivial solutio) x 3x x 0 v) 4 x 4x x 0 3 x x x x x x x 0 3 4

24 As : (Oly trivial solutio) vi) 3w x 0 u 4w 3x 0 u 3 w x 0 4 4u 3 5w 4x 0 As : u 7s 5 t, 6s 4 t, w s, x t Systems of liear equatios : ( poits to be remembered ). Determie whether the equatio 5x 7y 8yz 6 is liear. Solutio : No, sice the product yz of two ukows is of secod degree.. Is the equatio x y ez log 5 liear? Solutio : Yes, sice, e,log 5are costats.. Property : Cosider the degeerate liear equatio: 0x 0x... 0x b i) If the equatio s costat b 0, the the equatio has o solutio. ii) If the costat b 0, every vector i is a solutio.. Property : Cosider the liear equatio ax b. i) If a 0, the x b / a is the uique solutio. ii) If a 0, b 0, there is o solutio. iii) IF a 0, b 0, every scalar k is a solutio. Property A system of liear equatios, AX B has a solutio of ad oly of the rak of the coefficiet matrix is equal to the rak of its augmeted matrix.

25 5 MATRICES AND MATRIX OPERATIONS Uit structure.0 Objectives. Itroductio. Multiplicatio of Matrices.3 Matrices ad Liear Equatios.4 Traspose of a Matrix.5 Diagoal.0 OBJECTIVES : Rectagular arrays of real umbers arise i may cotexts other tha as augmeted matrix for systems of liear equatios. I this sectio we cosider such arrays as objects i their ow right ad develop some of their properties for use i our later work.. INTRODUCTION : Rectagular arrays of real umbers arise i may cotexts other tha as augmeted matrices for systems of liear equatios. I this sectio we cosider such arrays as objects i their ow right ad develop some of their properties for use i our later work. I this sectio, we defie elemetary row ad colum operatios for a matrix A. They are also kow by the commo ame of elemetary trasformatios of the matrix A. Matrices : A matrix is a rectagular array of umbers, real or complex. A m matrix is a rectagular array of m umbers arraged i m rows ad colums. Thus, a a a a a a am am a m is a m (real m by ) matrix, usually deoted by a sigle letter A. Each of the m umbers a, a,... is called a elemet of the matrix. The

26 6 elemet a ij occurs i the i th row ad j th colum. The matrix is deoted by A a ij. Traspose of a matrix : If A = aij m matrix B bji b a, the traspose of A is the m where m ji ij for i m, j. a a a am t If A the A am a m a a m The elemet a ij occurs i the i th ad j th colum. The matrix is deoted by A a ij. Symmetric Matrix : a ij A matrix A over is said to be symmetric if a for i, j. ji t A A. Skew symmetric Matrix : A matrix A over is said to be skew symmetric if t A A. Exercise : Let A be a matrix over. Show that i) ii) A A t A is symmetric. t A is skew symmetric. iii) A ca be expressed as a sum of a symmetric ad skew symmetric matrices. Exercise : If A ad B are symmetric matrices over, show that A B ad A are symmetric matrices over.. MULTIPLICATION OF MATRICES : Let A a ij be a m matrix over ad B b jk matrix. We defie the product, be a p AB c, i m, k p to be a mp matrix, where ik cik aijb jk for i m, k p. j

27 7 Note : The product AB may be defied but the product BA may ot be defied. However, if A, B are m matrices over, AB ad BA are both defied. However they may ot be equal. 0 For example, let A, 0 0 B 0 0, the 0 AB 0, 0 0 BA 0 0 0, AB BA. Note : A B, Mm are said to be equal. If aij bij for i m, j m m. However, we have associated law for multiplicatio of matrices. If AM m, B M p, C M pl. The, where A aij, B bij ABC ABC ad we also have distributive law. If AM, B,C M the m proof are obvious. We defie mp AM. A iductively for A B C AB AC A A. A, N,. Although, we itroduce matrices as a motivatio for the study of liear equatios. We are studyig matrices for their ow sake. The matrices form is a importat part of liear algebra. Idetity Matrix : I is called a idetity matrix. 0 0 Kroecker s delta : I δ, δ if i j ij ij Upper triagular matrix : 0 if i j A a ij is called a upper triagular matrix if aij 0 for i j a a a 0 a a A 0 0 a Lower triagular matrix : A aij is called a lower triagular matrix if 0 ij a for j i.

28 a 0 0 a a 0 A a a a Diagoal matrix : 8 A aij is called a diagoal matrix if 0 ij a a 0 A 0 0 a Scalar Matrix : a for i j A aij is called a scalar matrix if diagoal is c for some c. c c 0 A 0 0 c Ivertible Matrix : A aij B M R such that AB BA I. A ivertible matrix is also called a o-sigular matrix. The matrix B is called a iverse of A. is said to be a ivertible matrix if Theorem : If uique. Proof : If A M is a ivertible matrix, the iverse of A is AB BA I (i) AC CA I (ii) The BA I ( BA)C I.C ( post multiplyig by C ) Thus B(AC) = C C ( I = ) BI C (from ii) B C ( I = ) We shall deote the iverse of a ivertible matrix A by A. Propositio : If AM, BM the t t t AB B A m p.

29 Propositio : If A is a ivertible matrix over, the AT A T ivertible ad Proof : 9 AA I A A T T T AA I A A T T T T A.A I A A A T is ivertible ad AT A Propositio : If A,B M AB is ivertible ad T T A is are ivertible matrices, the the product AB B A. More geerally, if A,..., Ak are ivertible matrices the A A..., A A,..., A k k. Exercises. : 3 4 ) Let A 0 fid A, A, A. 0 0 a ) Let a,b ad let A b, B 3 fid AB, A, A. 0 0 Show that A is ivertible ad fid A. 0 3) i) Fid a matrix A such that A I ca you fid 0 all such matrices? ii) Fid a o-zero matrix A such that 4) Let A M of 5) Let A M of 3 A 0. A 0 show that I A is ivertible. 3 A A I 0 show that A is ivertible. 6) Let A, B, PM. If P is ivertible ad B P AP the show that B P A P for. A,BM of A, B are upper triagular matrices, fid if AB is upper triagular. cosθ si θ R θ, θ show that si θ cosθ Rθ Rθ Rθ θ for θ, θ. 7) 8) Let

30 ) Let A the fid ) Let A, B CA B? Justify your aswer. A? Is there a matrix C such that.3 MATRICES AND LINEAR EQUATIONS : We ow itroduce a algorithm that ca be used to solve a large system of equatio. This algorithm is called Gaussia Elimiatio method. We write the system of equatios as : aijx j b i, i m. (A) j x AX B, A a, X x m, B, A aij is called m x b m matrix. i.e. ij b the matrix of coefficiets ad the m a a b A, B the augmeted matrix. am am b m For example the system of liear equatios: x 3y 3z x 5y z 3 x 3 3 is deoted by matrix otatio as y 5. The 3 z 3 3 : augmeted matrix of the system is 5 : 3. Examples x 5y 7z 8w 7 3y 4w 8 is a system of 4 variables x y, z ad w. 3x 5z 9w 0 The system (A) ca be writte i the matrix form as

31 3 a a a x b a a a x b a a a x b m m m m m Agai this, we ca write as AX B; where A is a matrix of order m also it is called as matrix of coefficiet. X is a colum matrix of variables ad B is a colum matrix of costat. Summary : ) The m zero matrix deoted by 0m or simply 0, is the matrix whose elemets are all zero. Fid x, y, z, t. x y z 3 If 0 y 4 z w Solutio : x y 0 z 3 0 Set : y 4 0 z w 0 x 4, y 4, z 3, w 3 ) Show that for ay matrix A, we have A Solutio : A aij aij aij m m, A m, x y x 6 4 x y 3) Fi x, y, z, w if 3 z w w z w 3 Solutio : 3xx 4 3y x y 6 3z z w 3ww 3 x, y 4, z, w 3 Theorem : Let M be the collectio of a m matrices over a field K of scalars. The for ay matrices A a ij ay scalars K, K K. i) A B C A B C ii) A0 A iii) A A 0 iv) A B B A K A B K A K B v), B b ij, C c ij i M ad

32 3 K K A K A K A vi) vii) K K A K K A viii).a A Problem. Show that A A Aswer : Cosider A A A A 0.A 0 A 0 Thus, A A Or -A + A +(-)A = 0 - A Or A A ( - A + A = 0). Show that A A A, A A A 3A A A A A A A Thus A A A 3A A A A A A A 3 A A A A Matrix Multiplicatio : The product of row matrix ad a colum matrix with the same umber of elemets is their ier product as defied as : b a,..., a ab... ab akbk b k (5)(-) = 4+ (-8) 5 = 3 Example : A ad 0 4 B 3 6

33 33 Aswer : A & B 3 the AB is defied as AB 3 6 Now AB AB 4. Fid AB if A 0 5 & B Solutio : AB Fid Solutio : The first factor is ad the secod i.e Theorem Suppose that A, B, C are matrices ad K is scalar the AB C A BC associative law i) ii) AB C AB AC left distributive law iii) B C A BA CA right distributive law iv) K AB KA B AKB Let A, B 3 5 tha 6 6 AB Here AB BA i.e. matrix multiplicatio does ot obey the commutative law. A B C D AC AD BC BD 6. Show that Solutio : Usig distributive laws : A BC D A BC A B D AC BC AD BD AC AD BC BD

34 34.4 TRANSPOSE OF A MATRIX : The traspose of a matrix A, deoted by A T is the matrix obtaied by writig the rows of A, i order, as colums. If A aij, the A T = (a ij ) T 3, if m, A the 4 T A Theorem : The traspose operatios o matrices satisfieds. i) T T T A B A B ii) A T T T A T iii) KA KA K is scalar iv) T T T AB B. A.5 DIAGONAL : The diagoal of A a ij A is -square matrix. cosists of the elemets a, a,..., a where Trace of a -square matrix A a ij is the sum of its diagoal elemets i.e. tr A a a... a viz. trace of the matrix 3 A is Property : Suppose A a ij ad B b ij is ay scalar. The i) tr A B tra A trab, ii) tr ka k.tr A, iii) tr A.B tr B. A are -square matrices ad k Idetity matrix : I is the -square matrix with s o the diagoal ad 0s elsewhere. Examples :

35 35 0 I is the idetity matrix of order I3 0 0 is the idetity matrix of order 3 etc. 0 0 Kroecker delta : Kroecker delta is defied as 0 if i j ij if i j Accordigly, I ij D : Note: Trace of I Scalar Matrix k Dk K. I Example : Fid the scalar matrices of orders, 3 ad 4 correspodig to the scalar k 5. Solutio : Put 5s o the diagoal ad 0s elsewhere , 0 5 0, Powers of Matrices : The o-egative itegral powers of a square matrix M may be defied recursively by 0 r r M I, M M, M MM r,,... Property : p q p q i) A A A p q ii) If A ad B are commutative the A ad B are also commutative. Note : If A 0 ad K Sk 0 the ASK SK A SK. Note : A 0 Idempotet : A matrix E is idempotet if E E. Note : i) idetity matrix is idempotet I I ii) zero matrix is idempotet Example : Give matrix E 3 4 is idempotet. 3 p p Nilpotet : A is a ilpotet of class p if A 0 but A 0.

36 36 3 Example Show that A 5 6 is ilpotet of class Aswer : A A 3 A. A 0 Iverse Matrix : A square matrix A is ivertible if there exists a square matrix B such that AB BA I ; where I is idetity matrix. Note : A A 5 Example : Show that A 3 ad 3 5 B are iverses. 0 Aswer : AB I 0 0 BA I 0 a b Example : Whe is the geeral matrix A c d ivertible? What the is its iverse? Aswer : Take scalars x, y, z, t such that a b x y 0 c d z t 0 or ax bz ay bt 0 cx dz cy dt 0, ax bz ay bt 0 cx dz 0 cy dt Both of which have coefficiet matrix A. Set A ad bc. We kow that A is ivertible if A 0. I that case first ad secod system have the uique solutios. x d A, z c A, y b A, t a A. d A b A d b Thus A c A a A A c a I other words, whe A 0 the iverse of matrix A is obtaied by i) iterchagig the elemets or the mai diagoal, ii) takig the egative of the other elemets, ad

37 iii) multiplyig the matrix by A Example : Fid the iverse of A 3 Aswer : A 0, A exists. Next, iterchagig the diagoal elemets, take the egative of the other elemets ad multiply by A Example : Fid the iverse of A 4 Aswer : A, A exists. A. 3 3 A Example 3 : Fid the iverse of A 3 A Defiitio : A square matrix A of order A 0. Iverse of A is deoted ad defied by T A Traspose of A Adjoit of A A A Adjoit of A = Traspose of co-factor of A i j Co-factor of A = A Example : i) Fid iverse of the matrix Solutio : A A ii) If A 4 8 A A

38 38 5 iii) If A 0 A 0 A is ot ivertible. 3 5 iv) If A 3 Aswer : A 8 0, A exists. Take B = matrix of cofactors called as Adjoit of A. 3 3 B T B A T B 8 5 A v) Fid whether matrix is ivertible ad fid its iverse. 3 4 A 0 Solutio : A 4 0. B T B 5 3 A 5 4 A exists.

39 39

40 40 3 ELEMENTARY ROW OPERATIONS AND GAUSS-ELIMINATION METHOD, ROW ECHELON FORM Uit structure: 3.0 Objectives 3. Elemetary row operatios 3. Gauss Elimiatio Method to Solve AX=B 3.3 Matrix uits ad Elemetary Matrices 3.4 Elemetary Matrices 3.5 Liear Algebra System of Liear Equatios 3.0 OBJECTIVES : Object of this sectio is to develop a simple algorithm for fidig the iverse of a ivertible matrix. I this sectio, we give a systematic procedure for solvig of systems of liear equatios; it is based o the idea of reducig the augmeted matrix to a form that is simple eough so that the system of equatios ca be solved by ispectio. 3. ELEMENTARY ROW OPERATIONS: R R. E : Iterchage the i th row ad j th row : i j E : Multiply the i th row by a o-zero scalar K; Ri KRi, K 0. E 3 : Replace the i th row by K time the j th row plus the i th row: R KR R i j Explaatio : i a) Iterchagig the same two rows we obtai the origial matrix that is this operatio is its ow iverse b) Multiply the i th row by K ad the by K, or by K ad the by K, we obtai the origial matrix. I other words, the operatios i Ri KRi ad Ri K R are iverses.

41 4 c) Applyig the operatios Ri KR j Ri ad the operatio Ri KR j Ri, Elemetary Matrices : Let E be the matrix obtaied by applyig a elemetary row operatio to the idetity matrix I, i.e. let E e i. The E is called the elemetary matrix correspodig to the row operatios. Colum Operatios, Matrix equivalece : The elemetary colum operatios. F : Iterchage the i th colum ad the j th colum : Ci Cj F : Multiply the i th colum by a o-zero scalar K : KC C K 0 i i F 3 : Replace the i th colum by K times the j th colum plus the i th colum Ci KC j Ci We first defie elemetary row operatios o a m matrix over. The followig operatios o A operatios. i) Iterchagig th i ad ii) Multiplyig the Ri λri) are called elemetary row th j row of A deoted by Ri Rj. th i row by a o-zero scalar λ (deoted by iii) Addig a scalar multiple of i th row to R j λri+ R J. th j row of A deoted by Two m matrices are said to be row equivalet if oe ca be obtaied from the other by a successio of elemetary row operatios. Cosider the system of liear equatios. b b AX B, A aij B m bm If the augmeted matrix A,b is row equivalet to A, B, the the solutios of the system AX b are same as the solutios of the system A X B. Thus, to obtai the solutios of the system AX B, we try to reduce the matrix A, B to a simple form. Agai, our purpose is to elimiate ukows. We defie a row-echelo matrix :

42 4 Defiitio : A m matrix A a ij is called a row echelo matrix if A 0 or if there exists a iteger r. r mi m, ad itegers K K K r m s.t. for j K i i) For each i, i r, a ij 0 ii) For each i th row. iii) For each, a i, i r, a i 0 ik i, i r, a i 0 for s i. iv) aij 0 for all i r ad for all j. sk Note : Sometimes we cosider the pivot elemets are For example, A K, K 5, K 4, r 3. Theorem : Every matrix A, AM ik i is called the pivot elemet of is row equivalet to a matrix i row echelo form. 0 3 Example : Reduce the matrix 4 3 to row echelo form Aswer : A R R 0 3 (brigig the left most o-zero 3 etry to st row) 4 3 R3 R3 R (makig etries below st pivot 0)

43 R3 R3 R (makig etries below d pivot 0) Exercise : Reduce the followig matrices to row echelo form : i) ii) iii) iv) Note : I the system of equatios AX B, where A aij ad m x X th, the elemets i the j colum are coefficiets of the x m th ukow x j (or the j ukow) i each equatio of the system. We have already see that a system of o-homogeeous liear equatios may ot have a solutio. However, whe the system of liear equatios has a solutio, we fid the solutio by reducig the augmeted matrix to row echelo form. We assig arbitrary values to ukows correspodig to o-pivot elemets (if ay). The equatios are the solved by the method of back-substitutio. 3. GAUSS ELIMINATION METHOD TO SOLVE AX B : Algorithm : for augmeted matrix A, B. Step I Step II Step III : If the matrix cosists etirely of zeros. STOP. It is i rowechelo form. : Fid the first colum from the left cotaiig o-zero etry. If this is i i th row, ot is st row, perform R Ri. : Multiply that row by suitable umbers ad subtract from rows below it to make all etries below it 0.

44 44 Step IV : Repeat steps 3 o the matrix cosistig of remaiig rows. The process stops whe either o row remais at step 4 or all rows cosists of zeros. Call the leadig o-zero etries of each row pivots. If there are colums correspodig to o-pivot elemets, we assig arbitrary values to ukows correspodig to them ad we solve the system by back substitutio. The method breaks dow whe the pivots appear i last colum. Exercises. : i) Solve the give system of equatios by Gauss-Elimiatio method. x 3x 7x 5x x x x 4x 3x x x 4x x x x x x x ii) 3 4 x x x3 x4 x 7x 5x x x x x x 3 iii) 3 4 x x x3 x4 x x 3x x x x x x 3 4 x x x x 4 iv) 3 4 3x x 4x 3 4 x x 3x 5x x x 5x 6x x 8x 3x 4x v) 3 4 x 3x x x 3 4 x x x 0x x 5x x x 3 4 vi) 5x y z 4

45 45 x 3y z 30 x y 3z 5 As : x 39., y 6.7, z 8.97 vii) 0x 7y 3z 5 u 6 6x 8 y z 4u 5 3x y 4z u 5x 9y z 4u 7 As : u, z 7, y 4, x 5. Problems : Which of the followig matrices are i row echelo form. i) ii) iii) 0 iv) v) Problems : Reduce the followig matrices to row echelo form by a sequece of elemetary row operatios. i) ii)

46 46 4. Theory : Show that ay m matrix over ca be reduced to a matrix i row echelo form by performig a fiite umber of elemetary row operatios. 3.3 MATRIX UNITS AND ELEMENTARY MATRICES : Let rs place ad 0 elsewhere. I rs I deote a m matrix which has etry i the r, s th I rs are called matrix uits. Let A aij. The m I r,s rs a a ar ar A0 0 as as am a m Irr A A 0 0 as as 0 0 I.I 0, I.I I if r s rs rs rr rr rr r th row

47 47 s s rs sr I I A 0 0 a a ar a r 0 r th row s th row 3.4 ELEMENTARY MATRICES : We recall that there are three types of elemetary row operatios for matrices. i) Iterchagig two rows. ii) Multiplyig a row by a o-zero scalars. iii) Addig a scalar multiple of a row to aother row. The matrix E obtaied by performig ay of the elemetary row operatios o the idetity matrix is called a elemetary matrix. We shall deote i) The matrix obtaied by exchagig i th row ad j th row of idetity matrix by E ij. ii) The matrix obtaied by multiplyig i th λ 0 by Ei λ. iii) The matrix obtaied by addig matrix by Eij λ Theorem : Let A be a m real matrix. The, i) Eij ii) iii) row of idetity matrix by λ times j th row to i th of idetity A is the matrix obtaied by exchagig i th row ad j th row of A. Ei λ A is the matrix obtaied by multiplyig i th row of A by λ λ 0. Eij A. λ Ais the matrix obtaied by addig λ times j th row to i th row of Thus, multiplyig by a elemetary matrix is same as performig the correspodig elemetary matrix.. Propositio : A elemetary matrix is ivertible.

48 48 Propositio : Let A be a real matrix ad B is row equivalet to A, the A is ivertible if ad oly if B is ivertible. Proof : A is row equivalet to B. B is obtaied from A by a fiite sequece of elemetary row operatios. B E E E k ; where E E E k are elemetary matrices. If A is ivertible, B is product of ivertible matrices ad is ivertible (Each elemetary matrix is ivertible). If B is ivertible, the A E E... Ek B E k... E B. A is ivertible. Theorem : A matrix A is ivertible if ad oly if A is row equivalet to idetity matrix. (Ay upper triagular matrix with o-zero diagoal elemets is ivertible). Proof : If A is row equivalet to idetity matrix clearly A is ivertible, coversely, suppose A is ivertible, the the row echelo form of A does ot have ay row cosistig of zeros ad A is row equivalet to b b b 0 b b B 0 0 b b 0 for i ii The multiplyig i th equivalet to A), c c 0 c C 0 0 row by bii, we get a matrix C (row Ri Ci R gives Similarly, c

49 49 0 Ri Ci R makes th colum ad so o ad fially we get 0 idetity matrix. Thus, a ivertible matrix is row equivalet to idetity matrix. Further E k... E A I. i.e. A IE... Ek E... Ek which is a product of elemetary matrices. Exercises. :. For each of the followig elemetary matrices describe the correspodig elemetary row operatio ad write the iverse. i) 0 3 E ii) iii) E I each of the followig, fid a elemetary matrix E such that B EA. i) A, 3 B ii) iii) A 0, B 0 A 3, 3 B 3. Show that a ivertible matrix ca be expressed as a product of at most 4 elemetary matrices. 4. State which of the followig are elemetary matrices 0 0 i) 5 ii) 0 3

50 50 iii) 5 0 iv) v) vi) Express A as a product of elemetary matrices i) A ii) A Let A 5 E E A I. Gauss Elimiatio Method :, fid elemetary matrices E, E such that ) Solve the give system by Gauss Elimiatio Method. x 3x 7x 5x x x x 4x3 3x4 x5 x 4x3 x4 x5 3 x 5x 7x3 6x4 x5 7 Aswer : The augmeted matrix correspodig to the system is R R R R R R 3 3 R R R

51 5 R3 R3 6R R4 R4 7R R4 R4 R STOP The ukow correspodig to colums without pivots are x 3, x4. The solutio set is s t, s t,s,t, t,, 0,, 0 s,t i.e S,,,, S, t,,,, s,t 3.5 LINEAR ALGEBRA SYSTEM OF LINEAR EQUATIONS : Geeral form of the system of liear equatio is a x a x... a x b ax ax... ax b amx amx... amx bm (A) This is called as a system of m equatio i -ukow variables. Here a ij s ad b i s are real umber ad x, x,...,x are ukows aij s are called costats. Row Echelo Form : A matrix i row echelo form has zeros below each leadig. Example :

52 5 i) ii) iii) iv) Reduced row echelo form : A matrix i reduced row echelo form has zeros below ad above each leadig. Examples : i) ii) iii) iv) v) vi) Example : Suppose that the augmeted matrix for a system of liear equatios has bee reduced by row operatios to the give reduced row echelo form. Solve the system. a) b)

53 53 c) d) Solutio : (a) The correspodig system of equatios is x5 x x3 4 By ispectio (b) The correspodig system of equatio is x 4x 4 x x 6 4 x 3x 3 4 Sice x, x ad x 3 correspod to leadig s i the augmeted matrix, we call them leadig variables. Solvig for the leadig variables i terms of x 4 gives x 4x x x 4 6 x 4 3x 3 4 Sice x 4 ca be assiged a arbitrary value, say t, we have ifiitely may solutios. The solutio set is give by the formulas. x 4 t, x 6 t, x3 3 t, x4 t (c) x 4x5 6x x, 3 3x5, x4 5x5 x 4t 6 s, x s, x 3 t, x 5 t, x t x 0x 0x, therefore there is o solutio to the system. (d) 3 Elemetary matrices :

54 54 A matrix obtaied from a uit matrix I by meas of oe elemetary operatio o I is called a elemetary matrix. For example, Let I be a uit matrix of order 4. i.e. I the each of the followig is a elemetary matrix obtaied from I by the elemetary row operatios idicated : i) ii) iii) , by iterchages of R ad R 3., by 5R , by R 3R Exercises.3 : Fid rak ad iverses of the matrices if exists :

55 55 i) 4 3 A 0 3 As : A does ot exists. ii) 3 A 5 3, A iii) 6 4 A 4, 5 A does ot exists. iv) A, A 3 0 v) cosθ si θ 0 A si θ cosθ 0, 0 0 cosθ si θ 0 A si θ cosθ Iverse of a matrix A : a a a3 Let A a a a3 a 3 a3 a33 ad defied by be a 33 matrix. The iverse of A, deoted Adjoit ofa AdjA A A A T Adj A Cofactor of A ; T idicates traspose

56 56 Co-factor of A = Co-factor of each elemet of A i j a Co-factor of a Some correctios o pages os:, 9, 0, 7, 4, 30, 38, 4, 4, 43, 49, 57

57 57 4 VECTOR SPACES AND SUBSPACES Uit Structure: 4.0 Objectives 4. Itroductio 4.. Additio of vectors 4.. Scalar multiplicatio 4. Vector spaces 4.3 Subspace of a vector space 4.4 Summary 4.5 Uit Ed Exercise 4.0 OBJECTIVES This uit would make you to uderstad the followig cocepts : Vectors ad Scalars i plae ad space Additio ad scalar multiplicatio of vectors Various properties regardig additio ad scalar multiplicatio of vectors Idea of vector space Defiitio of vector space Various examples of vector space Defiitio of subspace Examples of subspace Results related to uio ad itersectio of subspace Liear Spa 4. INTRODUCTION The sigificat purpose of this uit is to study a vector space. A vector space is a collectio of vectors that satisfies a set of coditios. We ll look at may of the importat ideas that come with vector spaces oce we get the geeral defiitio of a vector ad a vector space. So, first we are goig to revise the cocept of ormal vectors, vector arithmetic ad their basic properties. We will see the vectors i 3 plae (space ) ad space (space ). Our fial aim is first to defie vector space usig the cocept of vectors ad scalars ad study various examples the to study some special subsets of vector space kow as subspaces with examples ad their properties.

58 58 Vectors : A vector ca be represeted geometrically by a directed lie segmet that starts at a poit A, called the iitial poit, ad eds at a poit B, called the termial poit. Below is a example of a vector i the plae. Remark : ) Vectors are deoted with a boldface lower case letter. For istace we could represet the vector above by v, w, a, b, etc. Also whe we ve explicitly give the iitial ad termial poits we will ofte represet the vector as, v = AB ) Vector has magitude ad directio. 3) I plae i.e. i IR we write vector v with iitial poit at origi ad termial poit at (x, y) as v = (x, y) ad a vector w with iitial poit at (x, y ) ad termial poit at (x, y ) as w = (x x, y y ). Similarly we ca express vectors i IR 3 (space). 4) Zero vector is a vector havig zero magitude. v is egative of vector v havig same magitude as that of v but opposite directio. 5) Scalars do ot have directio. All real umbers are scalars. Now we quickly discuss arithmetic of vectors. 4.. Additio of vectors : The additio of the vectors v ad w is show i the followig diagram. u v -v u + v u -v v u Suppose that we have two vectors v ad w the the differece of w from v, deoted by v - w is defied to be, v - w=v + (-w). 4.. Scalar multiplicatio: Suppose that v is a vector ad c is a o-zero scalar (i.e. c is a real umber) the the scalar multiple, cv, is the vector whose legth is c times the legth of v ad is i the directio of v if c is positive ad i the opposite directio of v if c is egative.