PART 2: DETERMINANTS, GENERAL VECTOR SPACES, AND MATRIX REPRESENTATIONS OF LINEAR TRANSFORMATIONS

Transcription

1 PART 2: DETERMINANTS, GENERAL VECTOR SPACES, AND MATRIX REPRESENTATIONS OF LINEAR TRANSFORMATIONS 3.1: THE DETERMINANT OF A MATRIX Learig Objectives 1. Fid the determiat of a 2 x 2 matrix 2. Fid the miors ad cofactors of a matrix 3. Use expasio by cofactors to fid the determiat of a matrix 4. Fid the determiat of a triagular matrix 5. Use elemetary row operatios to evaluate a determiat 6. Use elemetary colum operatios to evaluate a determiat 7. Recogize coditios that yield zero determiats Every matrix ca be associated with a real umber called its. Historically, the use of determiats arose from the recogitio of special that occur i the of systems of liear equatios. DEFINITION OF THE DETERMINANT OF A 2 x 2 MATRIX The of the matrix a A a det A. is give by **Note: I this text, ad are used iterchageably to represet the determiat of a matrix. I this cotext, the vertical bars are used to represet the of a matrix as opposed to the value. Example 1: Fid det A ad a. square solutios determiat 1 4 A 11 7 Ayazz detta absolute det B.. Araz IAI CREATED BY SHANNON MARTIN MYERS 75 a a b B 6 10 determiat patters determiat detca tdtd detcb a stfu41 B sail 3 tl g Yg

2 For 2 2 matrices Afiaa a A detata i DEFINITION OF MINORS AND COFACTORS OF A MATRIX d If A is a matrix, the the of the elemet is the determiat of the matrix obtaied by deletig the row ad the colum of A.The gig square mior Mij aij is give by C. ij Example 2: Fid the mior ad cofactor of a 12.ad b 23. D a a a A a a a a a a ith jth cofactor GD JMij b B b g y.dz Mz I TO td 3 L a23a3i q fy c 3D a M.i laag.ua z qaua33 Miz DEFINITION OF THE DETERMINANT OF A SQUARE MATRIX If A is a matrix of order 2, the the of the A is the of the etries i the first row of A multiplied by their respective. That is, det Czz C l iz _C 1 a.ziazz azz.az Czz square A A a1jc1. j1 j determiat Example 3: Cofirm that, for 2x2 matrices, this defiitio yields A a11a22 a21a12. A faaa detca ac ta zciz acta.zcizta.sc zt auf D't'a tak Dma cofactors taici sum CREATED BY SHANNON MARTIN MYERS 76 aazztaiz C am a Azz A zaz

3 Example 4: Fid B B THEOREM 3.1: DETERMINANT KIM OF A MATRIX PRODUCT If A be a square matrix of order. The the determiat of A is give by det det Est 1Bf b C t biz C z A A a C (ith row expasio) j1 i1 ij ij ij A A a C (jth colum expasio) ij Is there a easier way to complete the previous example? B t b D I 3 I to Dfc is tuk l C 67 t o 9 4 o HI Ai Ci ta iz Ci at tai Ci a j C j t a zjczjt Expasioby Cofactors tajcj i ol l C 6 O t 3 C 3 4 ly 21 CREATED BY SHANNON MARTIN MYERS

4 Alterative Method to evaluate the determiat of a 3 x 3 matrix: Copy the first ad secod colums of the matrix to form fourth ad fifth colums. The obtai the determiat by addig (or subtractig) the products of the six diagoals B T.E.at o o 1Bl 7 O 71 Example 5: Fid t t det A A tat 1 t t 1 8,1 o to o tat 1 7 3,9 0 to Z T IAI O IAI 1987 CREATED BY SHANNON MARTIN MYERS 78

5 What did you otice? determiat a triagular matrix the of like It looks Distheproductoftheetrieso the mai diagoal determiat is the mior each cofactors whe expadig by tha theorigial a matrix which is of order I less of THEOREM 3.2: DETERMINANT OF A TRIANGULAR MATRIX If A is a triagular matrix of order, the its determiat is the of the etries o product mai the. That is, det A diagoal Example 6: Fid the values of A. A aarrass, for which the determiat is zero M added fi f to fi Cosider the followig two matrices: A B Fid the determiat of each matrix. tgxiiiiyeg tat 6 to 4167g CREATED BY SHANNON MARTIN MYERS IBI 1801 b IBI.es A1 tsr3 p R3 madatig t.ieoeoiigrof RztRfsR3 to Ig 79

6 What did you fid out? F a relatioship betwee the determiats of a matrix A ad A i row echelo form leadig 1 s ot ecessary Take a closer look at the two matrices. Do you otice aythig? yes B is refla ad def B 5detCA THEOREM 3.3: ELEMENTARY ROW OPERATIONS AND DETERMINANTS Let A ad B be square matrices. Swappig 1. Whe B is obtaied from A by two of A,. 2. Whe B is obtaied from A by a of a row of A to aother row IBI IAI of A,. 3. Whe B is obtaied from A by a row of A by a c,. iterchagig rows lbl IAI addig multiple the Chagelig row is ot multipliedby aythig ozero multiplyig costat IBI CHI NOTE: Theorem 3.3 remais valid whe the word colum replaces the word row. Operatios performed o colums are called elemetary colum operatios. Example 7: Determie which property of determiats the equatio illustrates. a colum I colum 3 prop l b q prop 3 3 times CREATED BY SHANNON MARTIN MYERS 80

7 Example 8: Use elemetary row or colum operatios to fid the determiat of the matrix A RltR3 R3 ochageidet check Ak I3R2tR3 R3 o chage idef to o O 8 B detca detcb THEOREM 3.4: CONDITIONS THAT YIELD A ZERO DETERMINANT If A is a square matrix, ad ay oe of the followig coditios is true, the det A A etire row (or ) colum cosists of. Zeros rows colums equal 2. Two (or ) are. row colum multiple row colum 3. Oe (or ) is a of aother (or ). CREATED BY SHANNON MARTIN MYERS 81

8 Example 9: Prove the property. 1 a b 1 abc1, a 0, b 0, c 0. a b c c I iii fatal totti i Htt I it a felt b Itc Ita I tbt Ctbc D C b Ita btc tbc c b Ibltxtbc tab tactabc E abc ta t I t t l abc I l t tat f t Io I Itc t t i Itb B CREATED BY SHANNON MARTIN MYERS 82

9 3.2: PROPERTIES OF DETERMINANTS Learig Objectives 1. Fid the determiat of a matrix product ad a scalar multiple of a matrix 2. Fid the determiat of a iverse matrix ad recogize equivalet coditios for a osigular matrix 3. Fid the determiat of the traspose of a matrix 4. Use Cramer s Rule to solve a system of liear equatios 5. Use determiats to fid area, volume, ad equatios of lies ad plaes Example 1: Fid A, B, A B, A B, A B ad AB t t A B THEOREM 3.5: DETERMINANT OF A MATRIX PRODUCT If A ad B are square matrices of order, the t t AB.gg q9pgjatbft t 6 I 1 IAI IABt t.bz sft4l97sita 8E AI 2 o 2 1 Its IAB t IAI101 IBF ABA LABI101 IAHBt IOIMFAMB IAHBIt.rs 2Cit6 t3t3 D1A B1 if Ito 5 4 1B III µ B1 fits 5C 4767 LABI IAMBI 1AtBl AtHBl l4i So we kowthat IAIHBI 1AtBI CREATED BY SHANNON MARTIN MYERS 83

10 Example 2: Fid 3A ad 3B. 1 1 A B THEOREM 3.6: DETERMINANT OF A SCALAR MULTIPLE OF A MATRIX If A is a square matrix of order ad c is a scalar, the the determiat of ca is Proof: 3 3 t 2 2 1B IAI _ A Pg A zBl II tae A ED KAI 3213 BAKE IAI 2 2 KAI 1zBl 33lBl c Af LetA faai.ae adb EaaYcaz aij.cer 3 A Aau Araziad IBI Cca Gaza ca Ccaz IBI Ea aa Clara 1B IBt E aazz amaze EIAI.li CREATED BY SHANNON MARTIN MYERS 84

11 Example 3: Fid 3 6 A A, A, 1 B, ad B. 5 2 B 11 7 IAH R1t3R2 2 IA I IB I IBI 13 pot LIE B't EE E its i i.gs eipisoi i i i ivertible go 1B 1 1 is sigular IBI THEOREM 3.7: DETERMINANT OF AN INVERTIBLE MATRIX A square matrix A is ivertible (osigular) if ad oly if detca to Example 4: Fid 3 3 A A ad A. IAI IA't At CREATED BY SHANNON MARTIN MYERS 85

12 THEOREM 3.8: DETERMINANT OF AN INVERSE MATRIX If A is a ivertible matrix, the Proof: EQUIVALENT CONDITIONS FOR A NONSINGULAR MATRIX If A is a matrix, the the followig statemets are equivalet. 1. A is. 2. Ax 3. Ax b has a solutio for every colum matrix. 0 has oly the solutio. 4. A is to. 5. A ca be writte as the product of matrices. 6.. Example 5: Determie if the system of liear equatios has a uique solutio. x x x x x x x 2x 2x defca I detca sice A is ivertible A A I ad IAI 04hm3.7 IAA 1 1 Il tall A I i ad LAlIA'l 1 So 1A I L IAI ivertible uioteririae row equivalet IAI to I elemetary xt U 7 A gi detta IE 2 lifted FI fit 3 detla 10 0 IF to thesystem a uique solutio CREATED BY SHANNON MARTIN MYERS 86

13 Example 6: Fid T A ad A A 2 2 THEOREM 3.9: DETERMINANT OF A TRANSPOSE If A is a square matrix, the Example 7: Solve the system of liear equatios. Assume that a11a22 a21a12 0. a x a x b a x a x b CREATED BY SHANNON MARTIN MYERS 87

14 THEOREM 3.10: CRAMER S RULE If a system of liear equatios i variables has a coefficiet matrix A with a ozero determiat A, the the solutio of the system is Where the ith colum of Ai is the colum of costats i the system of equatios. Example 8: If possible, use Cramer s Rule to solve the system. a. x 2x x 4x b. 8x 7x 10x x 3 x 5 x x 9 x 2 x CREATED BY SHANNON MARTIN MYERS 88

15 AREA OF A TRIANGLE IN THE xy-plane The area of a triagle with vertices,,,, ad, x y x y x y is where the sig ( ) is chose to give positive area. Proof: CREATED BY SHANNON MARTIN MYERS 89

16 Example 9: Fid the area of the triagle whose vertices are 1, 1, 3, 5, ad 0, 2. TEST FOR COLLINEAR POINTS IN THE xy-plane Three poits,,,, ad, x y x y x y are colliear if ad oly if TWO-POINT FORM OF THE EQUATION OF A LINE A equatio of the lie passig through the distict poits x y ad, 1, 1 x y is give by 2 2 Example 10: Fid a equatio of the lie passig through the poits 4,7 ad 2, 4. CREATED BY SHANNON MARTIN MYERS 90

17 VOLUME OF A TETRAHEDRON The volume of a tetrahedro with vertices x y z, x, y, z, x, y, z, ad,, 1, 1, x y z is where the sig ( ) is chose to give positive volume. Example 11: Fid the volume of the tetrahedro with vertices 1,1,1, 0,0,0, 2,1, 1, ad 1,1, 2. TEST FOR COPLANAR POINTS IN SPACE Four poits, x1, y1, z 1, x, y, z,,, x y z, ad,, x y z are coplaar if ad oly if where the sig ( ) is chose to give positive volume. THREE-POINT FORM OF THE EQUATION OF A LINE A equatio of the plae passig through the distict poits by x y z,,, 1, 1, 1 x y z,ad,, x y z is give CREATED BY SHANNON MARTIN MYERS 91

18 3.3: GENERAL VECTOR SPACES Learig Objectives: 1. Determie whether a set of vectors is a vector space 2. Determie if a subset of a kow vector space V is a subspace of V 3. Write a vector as a liear combiatio of other vectors 4. Recogize bases i the vector spaces R, P, ad M m, 5. Determie whether a set S of vectors i a vector space V is a basis for V 6. Fid the dimesio of a vector space DEFINITION OF A VECTOR SPACE Let V be a set o which two operatios (vector additio ad scalar multiplicatio) are defied. If the listed axioms are satisfied for every u, v, ad w i V ad every scalar (real umber) c ad d, the V is called a vector space. Additio 1. u v is i V. uder additio 2. u v property 3. uvw property 4. V has a vector such that for every i V,. additive 5. For every i V, there is a vector i V deoted by such that. additive Scalar Multiplicatio 6. cu is i. uder scalar mult. 7. c uv property 8. cd u property 9. cd 10. u property 1 u idetity THEOREM 3.11: PROPERTIES OF SCALAR MULTIPLICATION Let v be ay elemet of a vector space V, ad let c be ay scalar. The the followig properties are true v 3. If, the or. 2. c v CREATED BY SHANNON MARTIN MYERS 92

19 Example 1: Determie whether the set, together with the idicated operatios, is a vector space. If it is ot, the idetify at least oe of the te vector space axioms that fails. a b a. The set of all 2 x 2 matrices of the form S : a, b, c, d R c 1. b. The set of all 2 x 2 osigular matrices with the stadard operatios. CREATED BY SHANNON MARTIN MYERS 93

20 IMPORTANT VECTOR SPACES CONTINUED = the set of all defied o the real lie. = the set of all defied o a. = the set of all. = the set of all of degree. = the set of all matrices. = the set of all matrices. Example 2: Describe the zero vector (the additive idetity) of the vector space. a. C, b. M1,4 Example 3: Describe the additive iverse of a vector i the vector space. a. C, b. M1,4 CREATED BY SHANNON MARTIN MYERS 94

21 Example 4: Determie whether the set of cotiuous fuctios, C, is a vector space. 1. Closure uder additio. 2. Commutativity uder additio. 3. Associativity uder additio. 4. Additive idetity. 5. Additive iverse. CREATED BY SHANNON MARTIN MYERS 95

22 6. Closure uder scalar multiplicatio. 7. Distributivity uder scalar multiplicatio (2 vectors ad 1 scalar). 8. Distributivity uder scalar multiplicatio (2 scalars ad 1 vector). 9. Associativity uder scalar multiplicatio. 10. Scalar multiplicative idetity. Coclusio? CREATED BY SHANNON MARTIN MYERS 96

23 Example 5: Determie whether the set W is a subspace of the vector space V with the stadard operatios of additio ad scalar multiplicatio. a. W : C 1,1 V : The set of all fuctios that are differetiable o 1,1 b. W : The set of all egative fuctios: f x 0. V : C, CREATED BY SHANNON MARTIN MYERS 97

24 W The set of all odd fuctios: f x f x V : C, c. :. d. W : The set of all x diagoal matrices. V : M : Z, e. W : The set of all x matrices whose trace is ozero. V : M : Z, CREATED BY SHANNON MARTIN MYERS 98

25 f. W : ax b : a, b R, a 0 V : C,. g. W : a 0 a :, 0 T ar a V : M : m, Z m, CREATED BY SHANNON MARTIN MYERS 99

26 Example 6: For the matrices 2 3 A 4 1 ad 0 5 B 1 2 i M 2,2, determie whether the give matrix is a liear combiatio of A ad B CREATED BY SHANNON MARTIN MYERS 100

27 Example 7: Determie whether the set of vectors i P2 is liearly idepedet or liearly depedet. 2 2 S x, x 1 Example 8: Determie whether the set of vectors i M 2, S,, is liearly idepedet or liearly depedet. CREATED BY SHANNON MARTIN MYERS 101

28 Example 9: Write the stadard basis for the vector space. M a. 5,2 b. P 3 Example 10: Determie whether S is a basis for the idicated vector space S 4 t t,5 t,3t 5, 2t 3t for P 3 CREATED BY SHANNON MARTIN MYERS 102

29 Example 11: Fid a basis for the vector space of all 3 x 3 symmetric matrices. What is the dimesio of this vector space? Example 11: Let T be the liear trasformatio from 2 1 P ito R give by the itegral T p p x dx. Fid the preimage of 1. That is, fid the polyomial fuctio(s) of degree 2 or less such that T p 1. 0 CREATED BY SHANNON MARTIN MYERS 103

30 3.4: RANK/NULLITY OF A MATRIX, SYSTEMS OF LINEAR EQUATIONS. AND COORDINATE VECTORS Learig Objectives: 1. Fid a basis for the row space, a basis for the colum space, ad the rak of a matrix 2. Fid the ullspace of a matrix 3. Fid a coordiate matrix relative to a basis i 4. Fid the trasitio matrix from the basis B to the basis B i 5. Represet coordiates i geeral -dimesioal spaces Let s do our math stretches! Cosider the followig matrix. R R A The row vectors of A are: The colum vectors of A are: DEFINITION OF ROW SPACE AND COLUMN SPACE OF A MATRIX Let A be a m matrix. The space of A is the of R by the vectors of A. The space of A is the subspace of A. R by the vectors of CREATED BY SHANNON MARTIN MYERS 104

31 Recall that two matrices are row-equivalet whe oe ca be obtaied from the other by operatios. THEOREM 3.12: ROW-EQUIVALENT MATRICES HAVE THE SAME ROW SPACE If a m matrix A is row-equivalet to a m matrix B, the the row space of A is equal to the row space of B. Proof: THEOREM 3.12: BASIS FOR THE ROW SPACE OF A MATRIX If a matrix A is row-equivalet to a matrix B i row-echelo form, the the ozero row vectors of B form a for the row space of A. To fid a basis for the row space of a matrix: reduce the matrix. The rows i the matrix are a for the row space of the matrix. Your aswer should be i the form of a of vectors. To fid a basis for the colum space of a matrix: Method 1: Use the steps above o the traspose of the matrix. Your aswer should be i the form of a of vectors. Method 2: Use reduced form of the origial matrix to fid the colums which cotai the (leadig ). Use the correspodig colums from the matrix for a basis. Your aswer should be i the form of a of vectors. CREATED BY SHANNON MARTIN MYERS 105

32 Example 1: Fid a basis for the row space ad colum space of the followig matrix: A Example 2: Fid a basis for the row space ad colum space of the followig matrix: A CREATED BY SHANNON MARTIN MYERS 106

33 THEOREM 3.13: ROW AND COLUMN SPACES HAVE EQUAL DIMENSIONS If A is a m matrix, the the row space ad the colum space of A have the same. DEFINITION OF THE RANK OF A MATRIX The of the (or ) space of a matrix A is called the of A ad is deoted by. Example 3: Fid the rak of the matrix from a. Example 1 b. Example 2 THEOREM 3.14: SOLUTIONS OF A HOMOGENEOUS SYSTEM If A is a m matrix, the the set of all solutios of the homogeeous system of liear equatios is a of called the of ad is deoted. So, The of the ullspace of A is called the of. Proof: CREATED BY SHANNON MARTIN MYERS 107

34 Example 4: Fid the ullspace of the followig matrix A, ad determie the ullity of A A THEOREM 3.15: DIMENSION OF THE SOLUTION SPACE If A is a m matrix of rak, the the of the solutio space of is. That is, Example 5: cosider the followig homogeeous system of liear equatios: x y 0 x y 0 a. Fid a basis for the solutio space. CREATED BY SHANNON MARTIN MYERS 108

35 b. Fid the dimesio of the solutio space. c. Fid the solutio of a cosistet system Ax b i the form xp x h THEOREM 3.16: SOLUTIONS OF A NONHOMOGENEOUS LINEAR SYSTEM If x is a particular solutio of the ohomogeeous system Ax b, the every solutio of this system ca p be writte i the form where xh is a solutio of the correspodig homogeeous system. Proof: THEOREM 3.17: SOLUTIONS OF A SYSTEM OF LINEAR EQUATIONS The system is cosistet if ad oly if is i the colum space of. Proof: CREATED BY SHANNON MARTIN MYERS 109

36 Example 7: cosider the followig ohomogeeous system of liear equatios: 2x4 y 5 z 8 7x14y4z 28 3 x 6 y z 12 Determie whether Ax b is cosistet. If the system is cosistet, write the solutio i the form x x p x h, where x p is a particular solutio of Ax b ad x h is a solutio of Ax 0. CREATED BY SHANNON MARTIN MYERS 110

37 COORDINATE REPRESENTATION RELATIVE TO A BASIS Let B v 1, v2,..., v be a ordered basis for a vector space V, ad let x be a vector i V such that The scalars c1, c2,..., c are called the of relative to the. The matrix (or coordiate ) of relative to is the matrix i whose are the coordiates of. Note: I, colum otatio is used for the coordiate matrix. For the vector, the are the coordiates of (relative to the for. So you have Example 8: Fid the coordiate matrix of x i x 1, 3, 0 R relative to the stadard basis. Example 9: Give the coordiate matrix of x relative to a (ostadard) basis B for coordiate matrix of x relative to the stadard basis. x B 4,0,7,3, 0,5, 1, 1, 3,4, 2,1, 0,1,5,0 B R, fid the CREATED BY SHANNON MARTIN MYERS 111

38 Example 10: Fid coordiate matrix of x i 6, 7, 4, 3, x 26,32 B R relative to the basis B. The matrix is called the from to, where is the coordiate matrix of relative to, ad is the coordiate matrix of relative to. Multiplicatio by the trasitio matrix chages a coordiate matrix relative to ito a coordiate matrix relative to. Chage of basis from to : Chage of basis from to : The chage of basis problem i example 3 ca be represeted by the matrix equatio: THEOREM 3.18: THE INVERSE OF A TRANSITION MATRIX If P is the trasitio matrix from a basis B to a basis B i matrix from to is give by. R, the is ivertible ad the trasitio CREATED BY SHANNON MARTIN MYERS 112

39 LEMMA Let B v v v ad 1, 2,..., v c u c u c u v c u c u c u v c u c u c u the the trasitio matrix from to is c11 c1 Q c 1 c B u1, u2,..., u be two bases for a vector space V. If THEOREM 3.19: TRANSITION MATRIX FROM B TO B Let B v v v ad,,..., 1 2 B u1, u2,..., u be two bases for to ca be foud usig Gauss-Jorda elimiatio o the 2 R. The the trasitio matrix from matrix B B as follows. Example 11: Fid the trasitio matrix from B to B. B 1,1, 1, 0, B 1, 0, 0,1 CREATED BY SHANNON MARTIN MYERS 113

40 Example 12: Fid the coordiate matrix of p relative to the stadard basis for P 3. p x x CREATED BY SHANNON MARTIN MYERS 114

41 3.5: THE KERNEL, RANGE, AND MATRIX REPRESENTATIONS OF LINEAR TRANSFORMATIONS, AND SIMILAR MATRICES Learig Objectives: 1. Fid the kerel of a liear trasformatio 2. Fid a basis for the rage, the rak, ad the ullity of a liear trasformatio 3. Determie whether a liear trasformatio is oe-to-oe or oto 4. Determie whether vector spaces are isomorphic 5. Fid the stadard matrix for a liear trasformatio 6. Fid the stadard matrix for the compositio of liear trasformatios ad fid the iverse of a ivertible liear trasformatio 7. Fid the matrix for a liear trasformatio relative to a ostadard basis 8. Fid ad use a matrix for a liear trasformatio 9. Show that two matrices are similar ad use the properties of similar matrices THE KERNEL OF A LINEAR TRANSFORMATION We kow from a earlier theorem that for ay liear trasformatio, the zero vector i maps to the vector i. That is,. I this sectio, we will cosider whether there are other vectors such that. The collectio of all such is called the of. Note that the zero vector is deoted by the symbol i both ad, eve though these two zero vectors are ofte differet. DEFINITION OF KERNEL OF A LINEAR TRANSFORMATION Let T : V W be a liear trasformatio. The the set of all vectors v i V that satisfy is called the of T ad is deoted by. Example 1: Fid the kerel of the liear trasformatio. a. T : R 3 R 3, T x, y, z x,0, z CREATED BY SHANNON MARTIN MYERS 115

42 T : P P, T a a xa x a x a 2a x 3a x b. c. T : P R, 2 1 T p p x dx 0 THEOREM 3.20: THE KERNEL IS A SUBSPACE OF V The kerel of a liear trasformatio T : V W is a subspace of the domai V. Proof: THEOREM 3.20: COROLLARY Let T : R R m be the liear trasformatio give by Tx kerel of T is equal to the solutio space of. Ax. The the CREATED BY SHANNON MARTIN MYERS 116

43 THEOREM 3.21: THE RANGE OF T IS A SUBSPACE OF W The rage of a liear trasformatio T : V W is a subspace of W. THEOREM 3.21: COROLLARY Let T : R R m be the liear trasformatio give by Tx equal to the of. Ax. The the colum space of is Example 2: Let T rage of T. 1 1 A v Av represet the liear trasformatio T. Fid a basis for the kerel of T ad the CREATED BY SHANNON MARTIN MYERS 117

44 DEFINITION OF RANK AND NULLITY OF A LINEAR TRANSFORMATION Let T : V W be a liear trasformatio. The dimesio of the kerel of T is called the of T ad is deoted by. The dimesio of the rage of T is called the of T ad is deoted by. THEOREM 3.22: SUM OF RANK AND NULLITY Let T : V W be a liear trasformatio from a -dimesioal vector space V ito a vector space W. The the of the of the ad is equal to the dimesio of the. That is, Proof: CREATED BY SHANNON MARTIN MYERS 118

45 Example 3: Defie the liear trasformatio T by T rak T A x Ax. Fid ker T, ullt, raget, ad 3 3 Example 4: Let T : R R be a liear trasformatio. Use the give iformatio to fid the ullity of T ad give a geometric descriptio of the kerel ad rage of T. T is the reflectio through the yz-coordiate plae: T x, y, z x, y, z CREATED BY SHANNON MARTIN MYERS 119

46 ONE-TO-ONE AND ONTO LINEAR TRANSFORMATIONS If the vector is the oly vector such that, the is. A fuctio is called oe-to-oe whe the of every i the rage cosists of a vector. This is equivalet to sayig that is oe-to-oe if ad oly if, for all ad i, implies that. THEOREM 3.23: ONE-TO-ONE LINEAR TRANSFORMATIONS Let T : V Proof: W be a liear trasformatio. The T is oe-to-oe if ad oly if. CREATED BY SHANNON MARTIN MYERS 120

47 THEOREM 3.24: LINEAR TRANSFORMATIONS Let T : V W be a liear trasformatio, where W is fiite dimesioal. The T is oto if ad oly if the of T is equal to the of W. Proof: THEOREM 3.25: ONE-TO-ONE AND ONTO LINEAR TRANSFORMATIONS Let T : V W be a liear trasformatio with vector spaces V ad W, of dimesio. The T is oe-to-oe if ad oly if it is. Example 5: Determie whether the liear trasformatio is oe-to-oe, oto, or either. 2 2 T : R R, T x, y x y, y x DEFINITION: ISOMORPHISM A liear trasformatio T : V W that is ad is called a. Moreover, if V ad W are vector spaces such that there exists a isomorphism from V to W, the V ad W are said to be to each other. CREATED BY SHANNON MARTIN MYERS 121

48 THEOREM 3.26: ISOMORPHIC SPACES AND DIMENSION Two fiite dimesioal vector spaces V ad W are if ad oly if they are of the same. Example 6: Determie a relatioship amog m,, j, ad k such that M m, is isomorphic to M jk,. WHICH FORMAT IS BETTER? WHY? Cosider T : R 3 R 3, T x, x, x 4x x 5 x, 2x x 6 x, x 3x ad 4 1 5x1 T x Ax x What do you thik? x The key to represetig a liear trasformatio by a matrix is to determie how it acts o a for. Oce you kow the of every vector i the, you ca use the properties of liear trasformatios to determie for ay i. Do you remember the stadard basis for R? Write this stadard basis for R i colum vector otatio. CREATED BY SHANNON MARTIN MYERS 122

49 B e1, e2,..., e THEOREM 3.26: STANDARD MATRIX FOR A LINEAR TRANSFORMATION m Let T : R R be a liear trasformatio such that, for the stadard basis vectors e i of a11 a12 a1 a 21 a 22 a 2 T e1, T e2,, T e, a a a the the m m1 m2 m matrix whose colums correspod to T e a11 a1 A am1 a m is such that T v Av for every v i i R. A is called the stadard matrix for T. R, Example 5: Fid the stadard matrix for the liear trasformatio T., 4,0,2 3 T x y x y x y CREATED BY SHANNON MARTIN MYERS 123

50 Example 2: Use the stadard matrix for the liear trasformatio Tto fid the image of the vector v.,,,2,2, v 3, 3 T x y x y x y x y Example 6: Cosider the followig liear trasformatio T: T is the reflectio through the yz-coordiate plae i R 3 : T x, y, z x, y, z, 2,3,4 a. Fid the stadard matrix A for the followig liear trasformatio T. v. b. Use A to fid the image of the vector v. c. Sketch the graph of v ad its image. CREATED BY SHANNON MARTIN MYERS 124

51 THEOREM 3.27: COMPOSITION OF LINEAR TRANSFORMATIONS Let : m T1 R R ad : m p T2 R R be liear trasformatios with stadard matrices A 1 ad A 2, respectively. The compositio T : R R p, defied by T T2 T1 stadard matrix A for T is give by the matrix product A A2A1. Proof: v v, is a liear trasformatio. Moreover, the Example 7: Fid the stadard matrices A ad A for T T2 T1 ad T T1 T T : R R, T x, y x, y, y T : R R, T x, y, z y, z CREATED BY SHANNON MARTIN MYERS 125

52 DEFINITION OF INVERSE LINEAR TRANSFORMATION If : T1 R R ad : T2 R R are liear trasformatios such that for every v i R, the T 2 is called the of T 1, ad T 1 is said to be. **Not every trasformatio has a. If is, however, the iverse is ad is deoted by. THEOREM 3.28 Let is T : R R be a liear trasformatio with a stadard matrix A. The the followig coditios are equivalet. T is. T is a. A is. If T is ivertible with stadard matrix A, the the stadard matrix for is. Example 8: Determie whether the liear trasformatio T x, y x y, x y its iverse. is ivertible. If it is, fid CREATED BY SHANNON MARTIN MYERS 126

53 THEOREM 3.29: TRANSFORMATION MATRIX FOR NONSTANDARD BASES Let V ad W be fiite-dimesioal vector spaces with bases B ad B, respectively, where B v1, v2,..., v. If T : V W is a liear trasformatio such that a11 a12 a1 a 21 a 22 a 2 T 1, T 2,, T v B v B v, B am1 am2 am the the m matrix whose colums correspod to T v 1B a11 a1 A am1 a m is such that for every i. T v by usig (a) the stadard matrix, ad (b) the matrix relative to B ad B. v 1,1,1, 1,1, 0, 0,1,1, B 1, 2, 1,1 Example 9: Fid T R R T x y z x y y z 3 2 :,,,,, 1,2,3, B CREATED BY SHANNON MARTIN MYERS 127

54 Example 10: Let B e x, xe x, x e x ad let be a basis for a subspace of W of the space of cotiuous fuctios, D x be the differetial operator o W. Fid the matrix for D x relative to the basis B. CREATED BY SHANNON MARTIN MYERS 128

55 A classical problem i liear algebra is determiig whether it is possible to fid a basis such that the matrix for relative to is. 1. Matrix for T relative to B : 2. Matrix for T relative to B : 3. Trasitio matrix from B to B : 4. Trasitio matrix from B to B : Example 11: Fid the matrix A relative to the basis B. 2 2 T : R R, T x, y x2 y,4 x, B 2,1, 1,1 CREATED BY SHANNON MARTIN MYERS 129

56 Example 12: Let B 1, 1, 2,1 ad 1,1, 1, 2 B be bases for 2 1 A 0 1 be the matrix for 2 2 T : R R relative to B. a. Fid the trasitio matrix P from B to B. 2 R, v B 1 4 T, ad let b. Use the matrices P ad A to fid v B ad T v B where B v 1 4 T. CREATED BY SHANNON MARTIN MYERS 130

57 DEFINITION OF SIMILAR MATRICES For square matrices A ad A of order, A is said to be similar to A whe there exists a ivertible 1 matrix P such that A P AP. THEOREM 3.30 Let A, B, ad C be square matrices of order. The the followig properties are true. 1. A is to. 2. If A is similar to B, the is to. 3. If A is similar to B ad B is similar to C, the is to. Proof: Example 13: Use the matrix P to show that A ad A are similar P 1 1 0, A 0 1 0, A CREATED BY SHANNON MARTIN MYERS 131

58 DIAGONAL MATRICES Diagoal matrices have may advatages over odiagoal matrices. d d 2 0 D k 0 0 D 0 0 d 0 0 Also, a diagoal matrix is its ow. Additioally, if all the diagoal elemets are ozero, the the iverse of a diagoal matrix is the matrix whose mai diagoal elemets are the of correspodig elemets i the origial matrix. Because of these advatages, it is importat to fid ways (if possible) to choose a basis for such that the matrix is Example 14: Suppose A is the matrix for T : R R relative to the stadard basis Fid the diagoal matrix A for T relative to the basis 1,1, 1, 1, 1,1, 1,1,1 B. CREATED BY SHANNON MARTIN MYERS 132

59 Example 15: Prove that if A is idempotet ad B is similar to A, the B is idempotet. (A matrix 2 is idempotet whe A A ). Proof: CREATED BY SHANNON MARTIN MYERS 133

60 4.1: INNER PRODUCT SPACES Learig Objectives: 1. Fid the legth of a vector ad fid a uit vector 2. Fid the distace betwee two vectors 3. Fid a dot product ad the agle betwee two vectors, determie orthogoality, ad verify the Cauchy-Schwartz Iequality, the triagle iequality, ad the Pythagorea Theorem 4. Use a matrix product to represet a dot product 5. Determie whether a fuctio defies a ier product, ad fid the ier product of two vectors i R, M m,, P, ad Ca, b 6. Fid a orthogoal projectio of a vector oto aother vector i a ier product space DEFINITION OF LENGTH OF A VECTOR IN The, or of a vector v v v v R,,..., 1 2 i is give by Whe would the legth of a vector equal to 0? Example 1: Cosider the followig vectors: 1 1 u 1, v 2, 2 2 a. Fid u CREATED BY SHANNON MARTIN MYERS 134

61 b. Fid v c. Fid u v d. Fid u v e. Fid 3u f. Fid 3 u Ay observatios? CREATED BY SHANNON MARTIN MYERS 135

62 THEOREM 4.1: LENGTH OF A SCALAR MULTIPLE Let v be a vector i R ad let c be a scalar. The where is the of c. Proof: THEOREM 4.2: UNIT VECTOR IN THE DIRECTION OF v If v is a ozero vector i R, the the vector has legth ad has the same as v. Proof: CREATED BY SHANNON MARTIN MYERS 136

63 Example 2: Fid the vector v with 3 v ad the same directio as 0, 2,1, 1 u. DEFINITION OF DISTANCE BETWEEN TWO VECTORS The distace betwee two vectors uad v i R is Example 3: Fid the distace betwee u 1,1, 2 ad 1, 3, 0 v. CREATED BY SHANNON MARTIN MYERS 137

64 DEFINITION OF THE ANGLE BETWEEN TWO VECTORS IN The betwee two ozero vectors i R R is give by CREATED BY SHANNON MARTIN MYERS 138

65 Example 4: Fid the agle betwee u 2, 1 ad 2,0 v. DEFINITION OF DOT PRODUCT IN R The dot product of u u u u ad v v v,,..., 1 2 v 1, 2,..., is the quatity CREATED BY SHANNON MARTIN MYERS 139

66 Example 5: Cosider the followig vectors: u 1, 2 v 2, 2 a. Fid u v b. Fid v v c. Fid u 2 d. Fid u v v e. Fid u5v CREATED BY SHANNON MARTIN MYERS 140

67 THEOREM 4.3: PROPERTIES OF THE DOT PRODUCT If u, v ad w are vectors i R, ad c is a scalar, the the followig properties are true. u v u v w c uv v v vv 0, ad vv 0 iff. Example 6: Fid 3 3 u v u v give that uu 8, uv 7, ad vv 6. THEOREM 4.4: THE CAUCHY-SCWARZ INEQUALITY If u ad v are vectors i R, the where deotes the value of Proof: u v. CREATED BY SHANNON MARTIN MYERS 141

68 Example 7: Verify the Cauch-Schwarz Iequality for u 1, 0 ad 1,1 v. CREATED BY SHANNON MARTIN MYERS 142

69 DEFINITION OF ORTHOGONAL VECTORS Two vectors u ad v i R are orthogoal if Example 7: Determie all vectors i 2 R that are orthogoal to 3,1 u. THEOREM 4.5: THE TRIANGLE INEQUALITY If u ad v are vectors i R, the Proof: CREATED BY SHANNON MARTIN MYERS 143

70 THEOREM 4.6: THE PYTHAGOREAN THEOREM If u ad v are vectors i R, the u ad v are orthogoal if ad oly if Example 8: Verify the Pythagore Theorem for the vectors u 3, 2 ad 4,6 v. DEFINITION OF AN INNER PRODUCT Let u, v, ad w be vectors i a vector space V, ad let c be ay scalar. A ier product o V is a fuctio that associates a real umber u, v with each pair of vectors u ad v ad satisfies the followig axioms. 1. uv, 2. u, v w 3. c u, v 4. v, v 0, ad v, v 0 iff NOTE: CREATED BY SHANNON MARTIN MYERS 144

71 Example 8: Show that the fuctio u, v uv 1 1 2uv 2 2uv 3 3 defies a ier product o u u, u, u ad v, v, v v R, where, Example 9: Show that the fuctio u, v uv 1 1uv 2 2uv 3 3 does ot defie a ier product o where, u u, u, u ad v, v, v v R, CREATED BY SHANNON MARTIN MYERS 145

72 THEOREM 4.7: PROPERTIES OF INNER PRODUCTS Let u, v, ad w be vectors i a ier product space V, ad let c be ay real umber. 1. 0v, 2. u v, w Proof: u, cv DEFINITION OF LENGTH, DISTANCE, AND ANGLE Let u ad v be vectors i a ier product space V. 1. The legth (or ) of u is. 2. The distace betwee u ad v is. 3. The agle betwee ad two vectors u ad v is give by. 4. u ad v are orthogoal whe. If, the u is called a vector. Moreover, if v is ay ozero vector i a ier product space V, the the vector is a vector ad is called the vector i the of v. Ier product o Ca, b is f, g. Ier product o M 2,2 is AB,. Ier product o P is pq, where ad. CREATED BY SHANNON MARTIN MYERS 146

73 Example 10: Cosider the followig ier product defied o u 0, 6, 1,1 a. Fid uv, v, ad uv, uv 1 12uv 2 2 R : b. Fid u c. Fid v d. Fid d u, v Example 11: Cosider the followig ier product defied: 1 2, f x x, f, g f x g x dx 1 a. Fid f, g g x x x 2 b. Fid f CREATED BY SHANNON MARTIN MYERS 147

74 c. Fid g d. Fid d f, g CREATED BY SHANNON MARTIN MYERS 148

75 THEOREM 4.8 Let u ad v be vectors i a ier product space V. Cauchy-Schwarz Iequality: Triagle Iequality: Pythagorea Theorem: u ad v are orthogoal if ad oly if Example 12: Verify the triagle iequality for AB, ab a b a b a b A 2 1, B 2 2, ad DEFINITION OF ORTHOGONAL PROJECTION Let u ad v be vectors i a ier product space V, such that oto v is v 0. The the orthogoal projectio of u CREATED BY SHANNON MARTIN MYERS 149

76 THEOREM 5.9: ORTHOGONAL PROJECTION AND DISTANCE Let u ad v be vectors i a ier product space V, such that v 0. The Example 13: Cosider the vectors u 1, 2 ad v 4, 2 a. proj v u. Use the Euclidea ier product to fid the followig: b. proj u v c. Sketch the graph of both proj v u ad proj u v. CREATED BY SHANNON MARTIN MYERS 150

77 4.2: ORTHONORMAL BASES: GRAM-SCHMIDT PROCESS Learig Objectives: 1. Show that a set of vectors is orthogoal ad forms a orthoormal basis, ad represet a vector relative to a orthoormal basis 2. Apply the Gram-Schmidt orthoormalizatio process 3 Cosider the stadard basis for R, which is This set is the stadard basis because it has useful characteristics such as The three vectors i the basis are, ad they are each. DEFINITIONS OF ORTHOGONAL AND ORTHONORMAL SETS A set S of a vector space V is called orthogoal whe every pair of vectors i S is orthogoal. If, i additio, each vector i the set is a uit vector, the S is called. For S v1, v2,..., v, this defiitio has the followig form. ORTHOGONAL ORTHONORMAL If is a, the it is a basis or a basis, respectively. THEOREM 4.10: ORTHOGONAL SETS ARE LINEARLY INDEPENDENT S v1, v2,..., v is a orthogoal set of vectors i a ier product space V, the S is liearly idepedet. Proof: If CREATED BY SHANNON MARTIN MYERS 151

78 THEOREM 4.10: COROLLARY If V is a ier product space of dimesio, the ay orthogoal set of ozero vectors is a basis for V. Example 1: Cosider the followig set i 4 R ,0,0,, 0,0,1,0, 0,1,0,0,,0,0, a. Determie whether the set of vectors is orthogoal. b. If the set is orthogoal, the determie whether it is also orthoormal. c. Determie whether the set is a basis for R. CREATED BY SHANNON MARTIN MYERS 152

79 THEOREM 4.11: COORDINATES RELATIVE TO AN ORTHONORMAL BASIS If B v1, v2,..., v is a orthoormal basis for a ier product space V, the the coordiate represetatio of a vector w relative to B is Proof: The coordiates of relative to the basis are called the coefficiets of relative to. The correspodig coordiate matrix of relative to is Example 2: Show that the set of vectors 2, 5, 10, 4 produce a orthoormal set. i 2 R is orthogoal ad ormalize the set to CREATED BY SHANNON MARTIN MYERS 153

80 Example 3: Fid the coordiate matrix of x 3, 4 relative to the orthoormal basis B,,, i 2 R. THEOREM 4.12: GRAM-SCHMIDT ORTHONORMALIZATION PROCESS Let 1, 2,..., Let w B v v v be a basis for a ier product V. B w1, w 2,..., w, where i v 1 1 v, w w v w w1, w1 v, w v, w w v w w w1, w1 w 2, w 2 w is give by v, w v, w v, w w v w w w w1, w1 w 2, w 2 w 1, w 1 Let u i w w i i B u1, u2,..., u is a orthoormal basis for V. Moreover,. The the set spa v, v,..., v spa u, u,..., u for k 1, 2,...,. 1 2 k 1 2 k CREATED BY SHANNON MARTIN MYERS 154

81 Example 4: Apply the Gram-Schmidt orthoormalizatio process to trasform the basis 1, 0, 0, 1,1,1, 1,1, 1 B for a subspace i product o 3 R ad use the vectors i the order they are give. 3 R ito a orthoormal basis. Use the Euclidea ier CREATED BY SHANNON MARTIN MYERS 155

82 4.3: MATHEMATICAL MODELS AND LEAST SQUARES ANALYSIS Learig Objectives: 1. Whe you are doe with your homework you should be able to 2. Defie the least squares problem 3. Fid the orthogoal complemet of a subspace ad the projectio of a vector oto a subspace 4. Fid the four fudametal subspaces of a matrix 5. Solve a least squares problem 6. Use least squares for mathematical modelig I this sectio we will study systems of liear equatios ad lear how to fid the of such a system. LEAST SQUARES PROBLEM Give a m matrix A ad a vector b i fid i m R, the problem is to m R such that is. DEFINITION OF ORTHOGONAL SUBSPACES The subspaces S 1 ad S 2 of R are orthogoal whe for all v 1 i S 1 ad v 2 i S 2. Example 1: Are the followig subspaces orthogoal? S spa 1, 0 ad S spa 1 1 CREATED BY SHANNON MARTIN MYERS 156

83 DEFINITION OF ORTHOGONAL COMPLEMENT If S is a subspace of R, the the orthogoal complemet of S is the set What s the orthogoal complemet of 0 i R? What s the orthogoal complemet of R? DEFINITION OF DIRECT SUM Let S 1 ad S 2 be two subspaces of R. If each vector ca be uiquely writte as the sum of a vector from ad a vector from,, the is the direct sum of ad, ad you ca write. Example 2: Fid the orthogoal complemet S, ad fid the direct sum S 0 1 S spa 1 1 S. CREATED BY SHANNON MARTIN MYERS 157

84 THEOREM 4.13: PROPERTIES OF ORTHOGONAL SUBSPACES Let S be a subspace of R, The the followig properties are true THEOREM 4.14: PROJECTION ONTO A SUBSPACE If u1, u2,..., u t is a orthoormal basis for the subspace S of R, ad v R, the Example 3: Fid the projectio of the vector v oto the subspace S S spa,, , v THEOREM 4.15: ORTHOGONAL PROJECTION AND DISTANCE Let S be a subspace of R ad let v R. The, for all u S, u proj S v, CREATED BY SHANNON MARTIN MYERS 158

85 FUNDAMENTAL SUBSPACES OF A MATRIX Recall that if A is a m matrix, the the colum space of A is a of cosistig of all vectors of the form,. The four fudametal subspaces of the matrix A are defied as follows. = ullspace of A = colum space of A = ullspace of T A = colum space of T A Example 4: Fid bases for the four fudametal subspaces of the matrix A CREATED BY SHANNON MARTIN MYERS 159

86 THEOREM 4.16: FUNDAMENTAL SUBSPACES OF A MATRIX If A is a m matrix, the ad are orthogoal subspaces of. ad are orthogoal subspaces of. SOLVING THE LEAST SQUARES PROBLEM Recall that we are attemptig to fid a vector x that miimizes, where A is a m matrix ad b is a vector i m R. Let S be the colum space of A :. Assume that b is ot i S, because otherwise the system Ax b would be. We are lookig for a vector i that is as close as possible to. This desired vector is the of oto. So, ad = is orthogoal to. However, this implies that is i, which equals. So, is i the of. The solutio of the least squares problem comes dow to solvig the liear system of equatios. These equatios are called the equatios of the least squares problem. CREATED BY SHANNON MARTIN MYERS 160

87 Example 5: Fid the least squares solutio of the system Ax b A 1, b CREATED BY SHANNON MARTIN MYERS 161

88 Example 6: The table shows the umbers of doctoral degrees y (i thousads) awarded i the Uited States from 2005 through Fid the least squares regressio lie for the data. The use the model to predict the umber of degrees awarded i Let t represet the year, with t = 5 correspodig to (Source: U.S. Natioal Ceter for Educatio Statistics) Year Doctoral Degrees, y CREATED BY SHANNON MARTIN MYERS 162

89 4.4: EIGENVALUES AND EIGENVECTORS, AND DIAGONALIZING MATRICES Learig Objectives: 1. Verify eigevalues ad correspodig eigevectors 2. Fid eigevectors ad correspodig eigespaces 3. Use the characteristic equatio to fid eigevalues ad eigevectors, ad fid the eigevalues ad eigevectors of a triagular matrix 4. Fid the eigevalues ad eigevectors of a liear trasformatio THE EIGENVALUE PROBLEM Oe of the most importat problems i liear algebra is the eigevalue problem. Whe A is a, do ozero vectors x i R exist such that Ax is a multiple of x? The scalar, deoted by ( ), is called a of the matrix A, ad the ozero vector x is called a of A correspodig to. DEFINITIONS OF EIGENVALUE AND EIGENVECTOR Let A be a matrix. The scalar is called a of A whe there is a vector x such that. The vector x is called a of A correspodig to. *Note that a eigevector caot be. Why ot? CREATED BY SHANNON MARTIN MYERS 163

90 Example 1: Determie whether x is a eigevector of A A 5 2 x 4, 4 a. b. x 8, 4 c. x 4,8 d. x 5, 3 THEOREM 4.17: EIGENVECTORS OF FORM A SUBSPACE If A is a matrix with a eigevalue, the the set of all eigevectors of, together with the zero vector is a subspace of R. This subspace is called the of. CREATED BY SHANNON MARTIN MYERS 164

91 Example 2: Fid the eigevalue(s) ad correspodig eigespace(s) of A. 1 k A 0 1 THEOREM 4.18: EIGENVALUES AND EIGENVECTORS OF A MATRIX Let A be a matrix. 1. A eigevalue of A is a scalar such that. 2. The eigevectors of A correspodig to are the solutios of. * The equatio is called the of A. Whe expaded to polyomial form, the polyomial is called the of A. This defiitio tells you that the of a matrix A correspod to the of the characteristic polyomial of A. CREATED BY SHANNON MARTIN MYERS 165

92 Example 3: Fid (a) the characteristic equatio ad (b) the eigevalues (ad correspodig eigevectos) of the matrix A THEOREM 4.19: EIGENVALUES OF TRIANGULAR MATRICES If A is a triagular matrix, the its eigevalues are the etries o its mai. Example 4: Fid the eigevalues of the triagular matrix CREATED BY SHANNON MARTIN MYERS 166

93 EIGENVALUES AND EIGENVECTORS OF LINEAR TRANSFORMATIONS A umber is called a of a liear trasformatio whe there is a vector such that. The vector x is called a of T correspodig to, ad the set of all eigevectors of (with the zero vector) is called the of. Example 6: Cosider the liear trasformatio T : R R whose matrix A relative to the stadard base is give. Fid (a) the eigevalues of A, (b) a basis for each of the correspodig eigespaces, ad (c) the matrix A for T relative to the basis B, where B is made up of the basis vectors foud i part b). 6 2 A 3 1 CREATED BY SHANNON MARTIN MYERS 167

94 4.5: DIAGONALIZATION Learig Objectives: 1. Fid the eigevectors of similar matrices, determie whether a matrix A is diagoalizable, ad fid a matrix P such that P 1 AP is diagoal 2. Fid, for a liear trasformatio T : V V is diagoal, a basis B for V such that the matrix for T relative to B DEFINITION OF A DIAGONALIZABLE MATRIX A matrix A is diagoalizable whe A is similar to a diagoal matrix. That is, A is diagoalizable whe there exists a ivertible matrix such that is a diagoal matrix. THEOREM 4.20: SIMILAR MATRICES HAVE THE SAME EIGENVALUES If A ad B are similar Proof: matrices, the the have the same. CREATED BY SHANNON MARTIN MYERS 168

95 Example 1: (a) verify that A is diagoalizable by computig ad Theorem 4.20 to fid the eigevalues of A A, P P 1 AP, ad (b) use the result of part (a) THEOREM 4.21: CONDITION FOR DIAGONALIZATION A matrix A is diagoalizable if ad oly if it has eigevectors. Proof: CREATED BY SHANNON MARTIN MYERS 169

96 Example 2: For the matrix A, fid, if possible, a osigular matrix P such that Verify P 1 AP A is a diagoal matrix with the eigevalues o the mai diagoal. P 1 APis diagoal. STEPS FOR DIAGONALIZING AN SQUARE MATRIX Let A be a matrix. 1. Fid liearly idepedet eigevectors for A (if possible) with correspodig eigevalues. If liearly idepedet eigevectors do ot exist, the A is ot diagoalizable. 2. Let P be the matrix whose colums cosist of these eigevectors. That is,. The diagoal matrix will have the eigevalues o its mai (ad elsewhere). Note that the order of the eigevectors used to form P will determie the order i which the eigevalues appear o the mai of. CREATED BY SHANNON MARTIN MYERS 170

97 THEOREM 4.22: SUFFICIENT CONDITION FOR DIAGONALIZATION If a matrix A has eigevalues, the the correspodig eigevectors are ad A is. Proof: Example 3: Fid the eigevalues of the matrix ad determie whether there is a sufficiet umber to guaratee that the matrix is diagoalizable Example 4: Fid a basis B for the domai of T such that the matrix for T relative to B is diagoal. 3 3 T : R R : T x, y, z 2x2y 3 z,2x y6 z, x 2y CREATED BY SHANNON MARTIN MYERS 171

98 4.5: SYMMETRIC MATRICES AND ORTHOGONAL DIAGONALIZATION Learig Objectives: 1. Recogize, ad apply properties of, symmetric matrices 2. Recogize, ad apply properties of, orthogoal matrices 3. Fid a orthogoal matrix P that orthogoally diagoalizes a symmetric matrix A SYMMETRIC MATRICES Symmetric matrices arise more ofte i tha ay other major class of matrices. The theory depeds o both ad. For most matrices, you eed to go through most of the diagoalizatio to ascertai whether a matrix is. We leared about oe exceptio, a matrix, which has etries o the mai. Aother type of matrix which is guarateed to be is a matrix. DEFINITION OF SYMMETRIC MATRIX A square matrix A is whe it is equal to its :. Example 1: Determie which of the matrices below are symmetric A 5 1, B , C , D CREATED BY SHANNON MARTIN MYERS 172

99 Example 2: Usig the diagoalizatio process, determie if A is diagoalizable. If so, diagoalize the matrix A. 6 1 A 1 5 CREATED BY SHANNON MARTIN MYERS 173

100 THEOREM 4.23: PROPERTIES OF SYMMETRIC MATRICES If A is a symmetric matrix, the the followig properties are true. 1. A is. 2. All of A are. 3. If is a of A with multiplicity, the has liearly eigevectors. That is, the of has dimesio. Proof of Property 1 (for a 2 x 2 symmetric matrix): Example 3: Prove that the symmetric matrix is diagoalizable. a a a A a a a a a a CREATED BY SHANNON MARTIN MYERS 174

101 Example 4: Fid the eigevalues of the symmetric matrix. For each eigevalue, fid the dimesio of the correspodig eigespace A CREATED BY SHANNON MARTIN MYERS 175

102 DEFINITION OF AN ORTHOGONAL MATRIX A square matrix P is whe it is ad whe. THEOREM 4.24: PROPERTY OF ORTHOGONAL MATRICES A matrix P is orthogoal if ad oly if its vectors form a set. Example 5: Determie whether the matrix is orthogoal. If the matrix is orthogoal, the show that the colum vectors of the matrix form a orthoormal set A CREATED BY SHANNON MARTIN MYERS 176

103 THEOREM 4.25: PROPERTY OF SYMMETRIC MATRICES Let A be a symmetric matrix. If 1 ad 2 are eigevalues of A, the their correspodig x 1 ad x 2 are. THEOREM 4.26: FUNDAMENTAL THEOREM OF SYMMETRIC MATRICES Let A be a matrix. The A is ad has eigevalues if ad oly if A is. Proof: STEPS FOR DIAGONALIZING A SYMMETRIC MATRIX Let A be a symmetric matrix. 1. Fid all of A ad determie the of each. 2. For eigevalue of multiplicity, fid a eigevector. That is, fid ay ad the it. 3. For eigevalue of multiplicity, fid a set of eigevectors. If this set is ot, apply the process. 4. The results of steps 2 ad 3 produce a set of eigevectors. Use these eigevectors to form the of. The matrix will be. The mai etries of are the of. CREATED BY SHANNON MARTIN MYERS 177

104 Example 5: Fid a matrix P such that proper diagoal form A T P AP orthogoally diagoalizes A. Verify that T P AP gives the Example 6: Prove that if a symmetric matrix A has oly oe eigevalue, the A I. CREATED BY SHANNON MARTIN MYERS 178

105 4.6: APPLICATIONS OF EIGENVALUES AND EIGENVECTORS Learig Objectives: 1. Fid the matrix of a quadratic form ad use the Pricipal Axes Theorem to perform a rotatio of axes for a coic ad a quadric QUADRATIC FORMS Every coic sectio i the xy-plae ca be writte as: If the equatio of the coic has o xy-term ( ), the the axes of the graphs are parallel to the coordiate axes. For secod-degree equatios that have a xy-term, it is helpful to first perform a of axes that elimiates the xy-term. The required rotatio agle is cot 2 a c. With b this rotatio, the stadard basis for. 2 R, is rotated to form the ew basis CREATED BY SHANNON MARTIN MYERS 179

106 Example 1: Fid the coordiates of a poit xy, i cos,si, si,cos B. 2 R relative to the basis ROTATION OF AXES The geeral secod-degree equatio ax bxy cy dx ey f 0 ca be writte i the form a x c y dxey f 0 by rotatig the coordiate axes couterclockwise through the agle a c, where is defied by cot 2. The coefficiets of the ew equatio are obtaied from the b substitutios x xcos ysi ad y xsi ycos. CREATED BY SHANNON MARTIN MYERS 180