Introduction to linear algebra/Lee W. Johnson, R. Dean Riess, Jimmy T. Arnold.-5th ed.课后习题答案
Ⅴ i CONTENTS 3.8 Least-Squares Solutions to Inconsistent Systems 3.9 Fitting Data and Least Squares Soluti 92 3.10 Supplementary Exercises 3.11C tual exercises 96 1 The Eigenvalue Problems 99 1.1 Introduction 4.2 Determinants and the Eigenvalue Problem ·· 101 4.3 Elementary Opcrations and Determinants 104 4.4 Eigenvalues and the Charactcristic Polynomial 108 4.5 Eigenvalues and eigenvectors .112 4.6 Complex eigenvalues and eigenvectors 117 4.7 Similarity Transformations Diagonalizatic 121 4.841 128 4. 9 Supplementary Exercises 132 4.10 Conceptual Exercises 132 5 Vector Spaces and Linear Transformations 135 5. 1 Introduction(No exercises) 135 5. 2 Vector Spaces 135 139 5. 4 Linear Independence, Bases, and Coordinates 144 5.5Di 147 5.6 Inner-products 150 5.7 Linear Transformations ..154 5. 8 Operations with Linear Transformations 158 5.9 Matrix Representations for Linear Transformations 161 5.10 Change of Basis and Diagonalization 166 5.11 Supplementary Exercises 171 5.12 Conceptual Exercises 173 6 Determinants 17 6. 1 Introduction(No exercises) 175 6.2 Cofactor Expansion of Determinants 175 6. 3 Elementary Operations and Determinants .178 6.4 Cramer's Rule 183 6.5 Applications of Determinants 186 6.6 Supplementary exercises 6.7 Conceptual Exercises 7 Eigenvalues and Applications 193 7. 1 Quadratic Fo 193 7. 2 Systems of Differential equations 197 7.3 Transformation to Hessenberg Form 199 7. 4 Eigenvalues of Hessenberg matrices 202 7.5 Householder Transformations 206 7.66 QR Factorization least-Squares 208 7.7 Matrix Polynomials &z The Cayley-Hamilton Theorem 7.8 Generalized Eigenvectors Diff. Eqns 212 7. 9 Supplementary Exercises 216 7.10 Conceptua.l Exercises C hapter 1 Matrices and Systems of Equations 1.1 Introduction to matrices and systems of Linear equations 1.L 2. Nonline 3. Linear 4. Nonl 5. Nonlinear 6. Linear 7. I3. 71+3·2 4 r 24·1 61-x2+ 146·2-(-1)+1=14 +2x2+43 42+2·(-1)+4·1 4 C1+x 0 1+(-1 0 3x1+4℃2 3:1+4(-1) 1+2 31+2·(1) 10.3 4.2 8 1. Unique solution 12. No Soluti 13. Infinitely many solutions 2 CHAPTER 1. MATRICES AND SYSTEMS OF EQUATIONS 14. No solution 15.(a) The planes do not intersect; that is, the planes are parallel (b)The planes intersect in a line or the planes are coincident 16. The planes intersect in the line t-(1-t)/2,=2, x-t 17. The planes intersect in the line T=4-3t, y=2t-1,i-t 18. Coincident planes 19.A 216 438 20.C 1271 243 1-3 21.O 2123 211 +2x2+7x3 2m1+2:2+4m3=3 23 21 + 6 x1+4x 3 4m1+3r2 821+ 31+2 11 111 4.4 B 11 113 25.A 11 1112 B 2 11111 20-11 11 26.A 0351 B 351 532 11 27.A一 131111 211 B 34-15 1 28.1= 123 5|.B= 111 123 52 1.1. INTRODUCTION TO MATRICES AND SYSTEMS OF LINEAR EQUATIONS 3 29.A=231|,B 2312 l-132 30. Elementary operations on equations: E2-2E1 Rcduccd systcm of cquations 2x1+32 6 Elementary row operations: R2- 2Rl Reduced augmented matrix 236 0-7-5 31. Lementary operations on equations: F2-F1, F3 +2FL 1+2℃2 Reduced syslem of equations +3 Elementary row operations: R2-R1, R3 +2 Rl 12-1 Reduced augmented matrix 100 5 132 6 32. Elementary operations on equations: E1+ E2, E3-2El 1 Reduced system of equations 4 4 Elementary row operations: R1< R2. R3-2R1 Reduced augmented matrix:0 114 0354 33. Elementary operations on equations: E2-E1, E3-3E1 1+ 9 Rcduccd systcm of cquations 2 21 Elementary row operations: R2- Rl, R3-3R1 11 Rcduccd augmented matrix:0 2 921 0-2-2 4 CHAPTER 1. MATRICES AND SYSTEMS OF EQUATIONS 34. Elementary operations on equations: 2+E1, E3 +2E1 x1+2 Reduced system of equations 3x2+3.x3-3x4 144 Elementary row operations: R2+R1, Rg+ 2R1 111-11 Reduced augmented matrix:02004 033-34 35. Lementary operations on equations: F2+ F1, F3+ Fl x1十2 Reduced system of equations 2+:3-4 3x2+6 Elementary row operations: R2++ Rl, R3+ Ri Reduced augmented matrix:0 1 1-13 03601 36. Elcmcntary opcrations on cquations: E2 E1, E3 3E1 1+C Reduced system of equations 000 Elementary row operations: R2-R1, R3 -3Rl 110 Reduced augmented matrix:0-2 0 0 37.(b)In each case, the graph of the resulting equation is a line 38. Now if u11=0 we easily obtain the equivalent system 1+ Thus we may suppose that a11+0. Then 111a122 E2(u21/1)E1 a211+a232=b2 l.1. INTRODUCTION TO MATRICES AND SYSTEMS OF LINEAR EQCATIONS 5 111a1 E2 (-21/(a1)a12+a2)x2=(-021/1)b1+b2 3 +a11b2 Each of al and(a11a22-a12a21)is non-zero 39.Le 11°1 b1 2 and let 11x1+ ca21C1+ca22C2-cb2 Suppose thal 1-51.J2=82 is d solutio on to A. Then a1151+01252-b1, and a2151+ But this means that ca21S1 Ca2282= cb2 and so C1=s1: 22= S2 is also a solution to B. Now suppose that m1=t1, 2=t2 is a solution to B. Then a11t1+a12t2=61 and ca21t1 + Ca22t2= cb2. Since c=0,a21I1+a2202=b2. 10. Let a111+ b1 x1+a22 nd let a1101+a12x2=b1 (a21+cu1)x1+(a22+c12)x2=b2+cbn Let 1-5 and z2- s2 be a solution to A. Then a1151+01282-b1 and a2151+022s2-b2 so a1151+a12S2-b1 and(a21+ca11)s1+(a22+Ca12)s2=b2+cb1 as required. Now if x1=t1 and 2=t2 is a solution to B thon alti t a12t2= b1 and (a21+ ca11)t1+(a22+ ca12)t2=b2+ cb1, sO a11t1+012t2=b1 and a21t1+a12t2=b2 as required 41. The proof is very similar to that of 45 and 46 42. By adding the two equations we obtain: 2. 1-2. 1 =4. Then m1=2 or 1=-1 and substituting these values in the second equation we find that there are three solutions 0 6 CHAPTER 1. MATRICES AND SYSTEMS OF EQUATIONS 1.2 Echelon form and gauss-Jordan elimination 1. The matrix is in echelon form. The row operation R2- 2R1 transforms the matrix to reduced echelon form 01 10-7 2. Echelon form. R2-2 Ri yields reduced row echelon form 3. Not in echelon form. (12) Ri, R2-1R1, (-1/5)R2 yields echelon form/1 3/21/2 2/5 4. Not in echelon form. R1 t R2 vields echelon form 011 Not in echelon form R1<>R2,(1/2)R1(1/2)H2 yiclds the cchclon form/I 01/22 0013/2 6. Not in echelon form 103/21/2 (1/2)Rl yields the echelon form 00 7. Not in echelon form. R2-4R3, R1-2Rg, R1-3R2 yields the reduced echelon form 1005 010-2 L00 1 1/232 Not in echelon form. (1/2)R1,(-1/3)Rs yields the echelon formo 12-1 9. Not in echelon form.(1/2) R2 yields the echelon form 000 1-43 6 10. Not in echelon form -Rl, (1/2)R2 yields the echelon form 0 1 1/2-3/2-3/2 000 0,x2=0. 12. The system is inconsistent 13. 1=-2+533 x2=l-3x3, I3 is arbitrary. 14.x1=1-2x3x2=0