Steven J. Leon University of Massachusetts, Dartmouth

Size: px
Start display at page:

Download "Steven J. Leon University of Massachusetts, Dartmouth"

Transcription

1 INSTRUCTOR S SOLUTIONS MANUAL LINEAR ALGEBRA WITH APPLICATIONS NINTH EDITION Steven J. Leon University of Massachusetts, Dartmouth Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo

2 The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs. Reproduced by Pearson from electronic files supplied by the author. Copyright 205, 200, 2006 Pearson Education, Inc. Publishing as Pearson, 75 Arlington Street, Boston, MA 026. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. ISBN-3: ISBN-0: X OPM

3 Contents Preface v Matrices and Systems of Equations Systems of Linear Equations 2 Row Echelon Form 2 3 Matrix Arithmetic 3 4 Matrix Algebra 6 5 Elementary Matrices 2 6 Partitioned Matrices 7 MATLAB Exercises 20 Chapter Test A 22 Chapter Test B 24 2 Determinants 27 The Determinant of a Matrix 27 2 Properties of Determinants 30 3 Additional Topics and Applications 33 MATLAB Exercises 35 Chapter Test A 35 Chapter Test B 36 3 Vector Spaces 38 Definition and Examples 38 2 Subspaces 42 3 Linear Independence 47 4 Basis and Dimension 50 5 Change of Basis 52 6 Row Space and Column Space 52 MATLAB Exercises 59 Chapter Test A 60 Chapter Test B 62 4 Linear Transformations 66 Definition and Examples 66 2 Matrix Representations of Linear Transformations 69 3 Similarity 7 MATLAB Exercise 72 iii

4 iv Contents Chapter Test A 73 Chapter Test B 74 5 Orthogonality 76 The Scalar product in R n 76 2 Orthogonal Subspaces 78 3 Least Squares Problems 8 4 Inner Product Spaces 85 5 Orthonormal Sets 90 6 The Gram-Schmidt Process 98 7 Orthogonal Polynomials 00 MATLAB Exercises 03 Chapter Test A 04 Chapter Test B 05 6 Eigenvalues 09 Eigenvalues and Eigenvectors 09 2 Systems of Linear Differential Equations 4 3 Diagonalization 5 4 Hermitian Matrices 23 5 Singular Value Decomposition 30 6 Quadratic Forms 32 7 Positive Definite Matrices 35 8 Nonnegative Matrices 38 MATLAB Exercises 40 Chapter Test A 44 Chapter Test B 45 7 Numerical Linear Algebra 49 Floating-Point Numbers 49 2 Gaussian Elimination 50 3 Pivoting Strategies 5 4 Matrix Norms and Condition Numbers 52 5 Orthogonal Transformations 62 6 The Eigenvalue Problem 64 7 Least Squares Problems 68 MATLAB Exercises 7 Chapter Test A 72 Chapter Test B 73

5 Preface This solutions manual is designed to accompany the ninth edition of Linear Algebra with Applications by Steven J. Leon. The answers in this manual supplement those given in the answer key of the textbook. In addition, this manual contains the complete solutions to all of the nonroutine exercises in the book. At the end of each chapter of the textbook there are two chapter tests (A and B) and a section of computer exercises to be solved using MATLAB. The questions in each Chapter Test A are to be answered as either true or false. Although the true-false answers are given in the Answer Section of the textbook, students are required to explain or prove their answers. This manual includes explanations, proofs, and counterexamples for all Chapter Test A questions. The chapter tests labeled B contain problems similar to the exercises in the chapter. The answers to these problems are not given in the Answers to Selected Exercises Section of the textbook; however, they are provided in this manual. Complete solutions are given for all of the nonroutine Chapter Test B exercises. In the MATLAB exercises. most of the computations are straightforward. Consequently, they have not been included in this solutions manual. On the other hand, the text also includes questions related to the computations. The purpose of the questions is to emphasize the significance of the computations. The solutions manual does provide the answers to most of these questions. There are some questions for which it is not possible to provide a single answer. For example, some exercises involve randomly generated matrices. In these cases, the answers may depend on the particular random matrices that were generated. Steven J. Leon sleon@umassd.edu v

6

7 Chapter Matrices and Systems of Equations SYSTEMS OF LINEAR EQUATIONS (d) (a) 3x + 2x 2 = 8 x + 5x 2 = 7 (b) 5x 2x 2 + x 3 = 3 2x + 3x 2 4x 3 = 0 (c) 2x + x 2 + 4x 3 = 4x 2x 2 + 3x 3 = 4 5x + 2x 2 + 6x 2 =

8 2 Chapter Matrices and Systems of Equations (d) 4x 3x 2 + x 3 + 2x 4 = 4 3x + x 2 5x 3 + 6x 4 = 5 x + x 2 + 2x 3 + 4x 4 = 8 5x + x 2 + 3x 3 2x 4 = 7 9. Given the system m x + x 2 = b m 2 x + x 2 = b 2 one can eliminate the variable x 2 by subtracting the first row from the second. One then obtains the equivalent system m x + x 2 = b (m m 2 )x = b 2 b (a) If m m 2, then one can solve the second equation for x x = b 2 b m m 2 One can then plug this value of x into the first equation and solve for x 2. Thus, if m m 2, there will be a unique ordered pair (x, x 2 ) that satisfies the two equations. (b) If m = m 2, then the x term drops out in the second equation 0 = b 2 b This is possible if and only if b = b 2. (c) If m m 2, then the two equations represent lines in the plane with different slopes. Two nonparallel lines intersect in a point. That point will be the unique solution to the system. If m = m 2 and b = b 2, then both equations represent the same line and consequently every point on that line will satisfy both equations. If m = m 2 and b b 2, then the equations represent parallel lines. Since parallel lines do not intersect, there is no point on both lines and hence no solution to the system. 0. The system must be consistent since (0, 0) is a solution.. A linear equation in 3 unknowns represents a plane in three space. The solution set to a 3 3 linear system would be the set of all points that lie on all three planes. If the planes are parallel or one plane is parallel to the line of intersection of the other two, then the solution set will be empty. The three equations could represent the same plane or the three planes could all intersect in a line. In either case the solution set will contain infinitely many points. If the three planes intersect in a point, then the solution set will contain only that point. 2 ROW ECHELON FORM 2. (b) The system is consistent with a unique solution (4, ). 4. (b) x and x 3 are lead variables and x 2 is a free variable. (d) x and x 3 are lead variables and x 2 and x 4 are free variables. (f) x 2 and x 3 are lead variables and x is a free variable. 5. (l) The solution is (0,.5, 3.5). 6. (c) The solution set consists of all ordered triples of the form (0, α, α). 7. A homogeneous linear equation in 3 unknowns corresponds to a plane that passes through the origin in 3-space. Two such equations would correspond to two planes through the origin. If one equation is a multiple of the other, then both represent the same plane through the origin and every point on that plane will be a solution to the system. If one equation is not a multiple of the other, then we have two distinct planes that intersect in a line through the

9 Section 3 Matrix Arithmetic 3 origin. Every point on the line of intersection will be a solution to the linear system. So in either case the system must have infinitely many solutions. In the case of a nonhomogeneous 2 3 linear system, the equations correspond to planes that do not both pass through the origin. If one equation is a multiple of the other, then both represent the same plane and there are infinitely many solutions. If the equations represent planes that are parallel, then they do not intersect and hence the system will not have any solutions. If the equations represent distinct planes that are not parallel, then they must intersect in a line and hence there will be infinitely many solutions. So the only possibilities for a nonhomogeneous 2 3 linear system are 0 or infinitely many solutions. 9. (a) Since the system is homogeneous it must be consistent. 3. A homogeneous system is always consistent since it has the trivial solution (0,..., 0). If the reduced row echelon form of the coefficient matrix involves free variables, then there will be infinitely many solutions. If there are no free variables, then the trivial solution will be the only solution. 4. A nonhomogeneous system could be inconsistent in which case there would be no solutions. If the system is consistent and underdetermined, then there will be free variables and this would imply that we will have infinitely many solutions. 6. At each intersection, the number of vehicles entering must equal the number of vehicles leaving in order for the traffic to flow. This condition leads to the following system of equations x + a = x 2 + b x 2 + a 2 = x 3 + b 2 x 3 + a 3 = x 4 + b 3 If we add all four equations, we get x 4 + a 4 = x + b 4 and hence x + x 2 + x 3 + x 4 + a + a 2 + a 3 + a 4 = x + x 2 + x 3 + x 4 + b + b 2 + b 3 + b 4 7. If (c, c 2 ) is a solution, then a + a 2 + a 3 + a 4 = b + b 2 + b 3 + b 4 a c + a 2 c 2 = 0 a 2 c + a 22 c 2 = 0 Multiplying both equations through by α, one obtains a (αc ) + a 2 (αc 2 ) = α 0 = 0 a 2 (αc ) + a 22 (αc 2 ) = α 0 = 0 Thus (αc, αc 2 ) is also a solution. 8. (a) If x 4 = 0, then x, x 2, and x 3 will all be 0. Thus if no glucose is produced, then there is no reaction. (0, 0, 0, 0) is the trivial solution in the sense that if there are no molecules of carbon dioxide and water, then there will be no reaction. (b) If we choose another value of x 4, say x 4 = 2, then we end up with solution x = 2, x 2 = 2, x 3 = 2, x 4 = 2. Note the ratios are still 6:6:6:. 3 MATRIX ARITHMETIC. (e)

10 4 Chapter Matrices and Systems of Equations (g) (d) (a) 5A = A + 3A = (b) 6A = = (2A) = = (c) A T = (A T ) T = 3 2 T 3 4 = 4 7 = A (a) A + B = = B + A 0 5 (b) 3(A + B) = = A + 3B = = (c) (A + B) T = T 5 0 = A T + B T = = (a) 3(AB) = 3 (3A)B = A(3B) = = = =

11 Section 3 Matrix Arithmetic T = (b) (AB) T = B T A T = = (a) (A + B) + C = = A + (B + C) = = (b) (AB)C = = A(BC) = = (c) A(B + C) = = AB + AC = = (d) (A + B)C = = AC + BC = = (b) x = (2, ) T is a solution since b = 2a +a 2. There are no other solutions since the echelon form of A is strictly triangular. (c) The solution to Ax = c is x = ( 5 2, 4 )T. Therefore c = 5 2 a 4 a 2.. The given information implies that 0 x = and x 2 = 0 are both solutions to the system. So the system is consistent and since there is more than one solution, the row echelon form of A must involve a free variable. A consistent system with a free variable has infinitely many solutions. 2. The system is consistent since x = (,,, ) T is a solution. The system can have at most 3 lead variables since A only has 3 rows. Therefore, there must be at least one free variable. A consistent system with a free variable has infinitely many solutions. 3. (a) It follows from the reduced row echelon form that the free variables are x 2, x 4, x 5. If we set x 2 = a, x 4 = b, x 5 = c, then x x 3 = 2 2a 3b c = 5 2b 4c and hence the solution consists of all vectors of the form x = ( 2 2a 3b c, a, 5 2b 4c, b, c) T (b) If we set the free variables equal to 0, then x 0 = ( 2, 0, 5, 0, 0) T is a solution to Ax = b and hence b = Ax 0 = 2a + 5a 3 = (8, 7,, 7) T

12 6 Chapter Matrices and Systems of Equations 4. If w 3 is the weight given to professional activities, then the weights for research and teaching should be w = 3w 3 and w 2 = 2w 3. Note that.5w 2 = 3w 3 = w, so the weight given to research is.5 times the weight given to teaching. Since the weights must all add up to, we have = w + w 2 + w 3 = 3w 3 + 2w 3 + w 3 = 6w 3 and hence it follows that w 3 = 6, w 2 = 3, w = 2. If C is the matrix in the example problem from the Analytic Hierarchy Process Application, then the rating vector r is computed by multiplying C times the weight vector w. r = Cw = = 5. A T is an n m matrix. Since A T has m columns and A has m rows, the multiplication A T A is possible. The multiplication AA T is possible since A has n columns and A T has n rows. 6. If A is skew-symmetric, then A T = A. Since the (j, j) entry of A T is a jj and the (j, j) entry of A is a jj, it follows that a jj = a jj for each j and hence the diagonal entries of A must all be The search vector is x = (, 0,, 0,, 0) T. The search result is given by the vector y = A T x = (, 2, 2,,, 2, ) T The ith entry of y is equal to the number of search words in the title of the ith book. 8. If α = a 2 /a, then 0 a a 2 = a a 2 = a a 2 α 0 b αa αa 2 + b a 2 αa 2 + b The product will equal A provided αa 2 + b = a Thus we must choose b = a 22 αa 2 = a 22 a 2a 2 a 4 MATRIX ALGEBRA. (a) (A + B) 2 = (A + B)(A + B) = (A + B)A + (A + B)B = A 2 + BA + AB + B 2 For real numbers, ab + ba = 2ab; however, with matrices AB + BA is generally not equal to 2AB. (b) (A + B)(A B) = (A + B)(A B) = (A + B)A (A + B)B = A 2 + BA AB B 2 For real numbers, ab ba = 0; however, with matrices AB BA is generally not equal to O.

13 Section 4 Matrix Algebra 7 2. If we replace a by A and b by the identity matrix, I, then both rules will work, since and (A + I) 2 = A 2 + IA + AI + B 2 = A 2 + AI + AI + B 2 = A 2 + 2AI + B 2 (A + I)(A I) = A 2 + IA AI I 2 = A 2 + A A I 2 = A 2 I 2 3. There are many possible choices for A and B. For example, one could choose A = 0 and B = More generally if A = a ca b cb B = db eb da ea then AB = O for any choice of the scalars a, b, c, d, e. 4. To construct nonzero matrices A, B, C with the desired properties, first find nonzero matrices C and D such that DC = O (see Exercise 3). Next, for any nonzero matrix A, set B = A+D. It follows that BC = (A + D)C = AC + DC = AC + O = AC 5. A 2 2 symmetric matrix is one of the form A = a b Thus b c A 2 = a2 + b 2 ab + bc ab + bc b 2 + c 2 If A 2 = O, then its diagonal entries must be 0. a 2 + b 2 = 0 and b 2 + c 2 = 0 Thus a = b = c = 0 and hence A = O. 6. Let D = (AB)C = a b + a 2 b 2 a b 2 + a 2 b 22 a 2 b + a 22 b 2 a 2 b 2 + a 22 b 22 It follows that If we set E = A(BC) = then it follows that d = (a b + a 2 b 2 )c + (a b 2 + a 2 b 22 )c 2 = a b c + a 2 b 2 c + a b 2 c 2 + a 2 b 22 c 2 d 2 = (a b + a 2 b 2 )c 2 + (a b 2 + a 2 b 22 )c 22 = a b c 2 + a 2 b 2 c 2 + a b 2 c 22 + a 2 b 22 c 22 d 2 = (a 2 b + a 22 b 2 )c + (a 2 b 2 + a 22 b 22 )c 2 = a 2 b c + a 22 b 2 c + a 2 b 2 c 2 + a 22 b 22 c 2 d 22 = (a 2 b + a 22 b 2 )c 2 + (a 2 b 2 + a 22 b 22 )c 22 = a 2 b c 2 + a 22 b 2 c 2 + a 2 b 2 c 22 + a 22 b 22 c 22 a a 2 a 2 a 22 c c 2 c 2 c 22 b c + b 2 c 2 b c 2 + b 2 c 22 b 2 c + b 22 c 2 b 2 c 2 + b 22 c 22 e = a (b c + b 2 c 2 ) + a 2 (b 2 c + b 22 c 2 ) = a b c + a b 2 c 2 + a 2 b 2 c + a 2 b 22 c 2

14 8 Chapter Matrices and Systems of Equations 9. Thus and hence and A 4 = O. If n > 4, then e 2 = a (b c 2 + b 2 c 22 ) + a 2 (b 2 c 2 + b 22 c 22 ) = a b c 2 + a b 2 c 22 + a 2 b 2 c 2 + a 2 b 22 c 22 e 2 = a 2 (b c + b 2 c 2 ) + a 22 (b 2 c + b 22 c 2 ) = a 2 b c + a 2 b 2 c 2 + a 22 b 2 c + a 22 b 22 c 2 e 22 = a 2 (b c 2 + b 2 c 22 ) + a 22 (b 2 c 2 + b 22 c 22 ) = a 2 b c 2 + a 2 b 2 c 22 + a 22 b 2 c 2 + a 22 b 22 c 22 d = e d 2 = e 2 d 2 = e 2 d 22 = e 22 (AB)C = D = E = A(BC) A = (a) The matrix C is symmetric since (b) The matrix D is symmetric since A = A n = A n 4 A 4 = A n 4 O = O (c) The matrix E = AB is not symmetric since and in general, AB BA. (d) The matrix F is symmetric since (e) The matrix G is symmetric since C T = (A + B) T = A T + B T = A + B = C D T = (AA) T = A T A T = A 2 = D E T = (AB) T = B T A T = BA F T = (ABA) T = A T B T A T = ABA = F G T = (AB + BA) T = (AB) T + (BA) T = B T A T + A T B T = BA + AB = G (f) The matrix H is not symmetric since H T = (AB BA) T = (AB) T (BA) T = B T A T A T B T = BA AB = H. (a) The matrix A is symmetric since (b) The matrix B is not symmetric since (c) The matrix D is symmetric since (d) The matrix E is symmetric since A T = (C + C T ) T = C T + (C T ) T = C T + C = A B T = (C C T ) T = C T (C T ) T = C T C = B E T A T = (C T C) T = C T (C T ) T = C T C = D = (C T C CC T ) T = (C T C) T (CC T ) T = C T (C T ) T (C T ) T C T = C T C CC T = E

15 Section 4 Matrix Algebra 9 (e) The matrix F is symmetric since F T = ((I + C)(I + C T )) T = (I + C T ) T (I + C) T = (I + C)(I + C T ) = F (e) The matrix G is not symmetric. F T F = (I + C)(I C T ) = I + C C T CC T = ((I + C)(I C T )) T = (I C T ) T (I + C) T = (I C)(I + C T ) = I C + C T CC T F and F T are not the same. The two middle terms C C T and C + C T do not agree. 2. If d = a a 22 a 2 a 2 0, then a 22 a 2 a a a 22 a 2 a 2 0 a 2 d = d a 2 a a 2 a 22 0 a a 22 a 2 a 2 = I d a [ a 2 a ] a a 22 a 2 a a 2 d = a 2 a 22 d a 2 a 0 a a 22 a 2 a 2 = I d Therefore a 22 a 2 = A d a 2 a 3. (b) If A were nonsingular and AB = A, then it would follow that A AB = A A and hence that B = I. So if B I, then A must be singular. 5. Since A A = AA = I it follows from the definition that A is nonsingular and its inverse is A. 6. Since it follows that A T (A ) T = (A A) T = I (A ) T A T = (AA ) T = I (A ) T = (A T ) 7. If Ax = Ay and x y, then A must be singular, for if A were nonsingular, then we could multiply by A and get 8. For m =, A Ax = A Ay x = y (A ) = A = (A ) Assume the result holds in the case m = k, that is, It follows that and (A k ) = (A ) k (A ) k+ A k+ = A (A ) k A k A = A A = I A k+ (A ) k+ = AA k (A ) k A = AA = I

16 0 Chapter Matrices and Systems of Equations Therefore (A ) k+ = (A k+ ) and the result follows by mathematical induction. 9. If A 2 = O, then (I + A)(I A) = I + A A + A 2 = I and (I A)(I + A) = I A + A + A 2 = I Therefore I A is nonsingular and (I A) = I + A. 20. If A k+ = O, then and (I + A + + A k )(I A) = (I + A + + A k ) (A + A A k+ ) = I A k+ = I (I A)(I + A + + A k ) = (I + A + + A k ) (A + A A k+ ) = I A k+ = I Therefore I A is nonsingular and (I A) = I + A + A A k. 2. Since R T cos θ sin θ R = cos θ sin θ = 0 sin θ cos θ sin θ cos θ and RR T = cos θ sin θ it follows that R is nonsingular and R = R T G 2 = H 2 = (I 2uu T ) 2 sin θ cos θ sin θ = 0 cos θ sin θ cos θ 0 cos2 θ + sin 2 θ 0 0 cos 2 θ + sin 2 θ = I 4uu T + 4uu T uu T = I 4uu T + 4u(u T u)u T = I = I 4uu T + 4uu T = I (since u T u = ) 24. In each case, if you square the given matrix, you will end up with the same matrix. 25. (a) If A 2 = A, then (I A) 2 = I 2A + A 2 = I 2A + A = I A (b) If A 2 = A, then (I 2 A)(I + A) = I 2 A + A 2 A2 = I 2 A + A 2 A = I and (I + A)(I 2 A) = I + A 2 A 2 A2 = I + A 2 A 2 A = I 26. (a) Therefore I + A is nonsingular and (I + A) = I 2 A. D 2 = d d d 2 nn

17 Section 4 Matrix Algebra Since each diagonal entry of D is equal to either 0 or, it follows that d 2 jj = d jj, for j =,..., n and hence D 2 = D. (b) If A = XDX, then A 2 = (XDX )(XDX ) = XD(X X)DX = XDX = A 27. If A is an involution, then A 2 = I and it follows that B 2 C 2 = 4 (I + A)2 = 4 (I + 2A + A2 ) = 4 (2I + 2A) = (I + A) = B 2 = 4 (I A)2 = 4 (I 2A + A2 ) = 4 (2I 2A) = (I A) = C 2 So B and C are both idempotent. BC = 4 (I + A)(I A) = 4 (I + A A A2 ) = (I + A A I) = O (A T A) T = A T (A T ) T = A T A (AA T ) T = (A T ) T A T = AA T 29. Let A and B be symmetric n n matrices. If (AB) T = AB, then 30. (a) Conversely, if BA = AB, then BA = B T A T = (AB) T = AB (AB) T = B T A T = BA = AB B T C T = (A + A T ) T = A T + (A T ) T = A T + A = B = (A A T ) T = A T (A T ) T = A T A = C (b) A = 2 (A + AT ) + 2 (A AT ) 34. False. For example, if A = 2 3, B = 4, x = then Ax = Bx = 5 5 however, A B. 35. False. For example, if A = 0 and B = then it is easy to see that both A and B must be singular, however, A + B = I, which is nonsingular. 36. True. If A and B are nonsingular, then their product AB must also be nonsingular. Using the result from Exercise 23, we have that (AB) T is nonsingular and ((AB) T ) = ((AB) ) T. It follows then that ((AB) T ) = ((AB) ) T = (B A ) T = (A ) T (B ) T

18 2 Chapter Matrices and Systems of Equations 5 ELEMENTARY MATRICES 2. (a) 0, type I 0 (b) The given matrix is not an elementary matrix. Its inverse is given by (c) (d) 5. (c) Since / , type III, type II C = F B = F EA where F and E are elementary matrices, it follows that C is row equivalent to A (b) E = 3 0, E 2 = 0 0, E 3 = The product L = E E 2 E 3 is lower triangular. 0 0 L = A can be reduced to the identity matrix using three row operations The elementary matrices corresponding to the three row operations are E = 0, E 3 2 =, E 0 3 = So and hence and A = E 3 E 2 E. 8. (b) (d) (a) E 3 E 2 E A = I A = E E 3 E 3 = =

19 (e) (b) XA + B = C X = (C B)A = (d) XA + C = X XA XI = C X(A I) = C X = C(A I) = = Section 5 Elementary Matrices (a) If E is an elementary matrix of type I or type II, then E is symmetric. Thus E T = E is an elementary matrix of the same type. If E is the elementary matrix of type III formed by adding α times the ith row of the identity matrix to the jth row, then E T is the elementary matrix of type III formed from the identity matrix by adding α times the jth row to the ith row. (b) In general, the product of two elementary matrices will not be an elementary matrix. Generally, the product of two elementary matrices will be a matrix formed from the identity matrix by the performance of two row operations. For example, if E = 2 0 and E 2 = then E and E 2 are elementary matrices, but 0 0 E E 2 = is not an elementary matrix. 4. If T = UR, then Since U and R are upper triangular If i > j, then t ij = n u ik r kj k= u i = u i2 = = u i,i = 0 r j+,j = r j+2,j = r nj = 0 t ij = = = 0 j u ik r kj + k= j 0 r kj + k= n k=j+ n k=j+ u ik r kj u ik 0

20 4 Chapter Matrices and Systems of Equations Therefore T is upper triangular. If i = j, then Therefore t jj = t ij = 5. If we set x = (2, 4) T, then = i u ik r kj + u jj r jj + k= i 0 r kj + u jj r jj + k= = u jj r jj t jj = u jj r jj n k=j+ n k=j+ j =,..., n Ax = 2a + a 2 4a 3 = 0 u ik r kj Thus x is a nonzero solution to the system Ax = 0. But if a homogeneous system has a nonzero solution, then it must have infinitely many solutions. In particular, if c is any scalar, then cx is also a solution to the system since A(cx) = cax = c0 = 0 Since Ax = 0 and x 0, it follows that the matrix A must be singular. (See Theorem.5.2) 6. If a = 3a 2 2a 3, then a 3a 2 + 2a 3 = 0 Therefore x = (, 3, 2) T is a nontrivial solution to Ax = 0. It follows from Theorem.5.2 that A must be singular. 7. If x 0 0 and Ax 0 = Bx 0, then Cx 0 = 0 and it follows from Theorem.5.2 that C must be singular. 8. If B is singular, then it follows from Theorem.5.2 that there exists a nonzero vector x such that Bx = 0. If C = AB, then Cx = ABx = A0 = 0 Thus, by Theorem.5.2, C must also be singular. 9. (a) If U is upper triangular with nonzero diagonal entries, then using row operation II, U can be transformed into an upper triangular matrix with s on the diagonal. Row operation III can then be used to eliminate all of the entries above the diagonal. Thus, U is row equivalent to I and hence is nonsingular. (b) The same row operations that were used to reduce U to the identity matrix will transform I into U. Row operation II applied to I will just change the values of the diagonal entries. When the row operation III steps referred to in part (a) are applied to a diagonal matrix, the entries above the diagonal are filled in. The resulting matrix, U, will be upper triangular. 20. Since A is nonsingular it is row equivalent to I. Hence, there exist elementary matrices E, E 2,..., E k such that E k E A = I It follows that and A = E k E E k E B = A B = C The same row operations that reduce A to I, will transform B to C. Therefore, the reduced row echelon form of (A B) will be (I C). u ik 0

21 Section 5 Elementary Matrices 5 2. (a) If the diagonal entries of D are α, α 2,..., α n and the diagonal entries of D 2 are β, β 2,..., β n, then D D 2 will be a diagonal matrix with diagonal entries α β,..., α n β n and D 2 D will be a diagonal matrix with diagonal entries β α, β 2 α 2,..., β n α n. Since the two have the same diagonal entries, it follows that D D 2 = D 2 D. (b) AB = A(a 0 I + a A + + a k A k ) = a 0 A + a A a k A k+ = (a 0 I + a A + + a k A k )A = BA 22. If A is symmetric and nonsingular, then (A ) T = (A ) T (AA ) = ((A ) T A T )A = A 23. If A is row equivalent to B, then there exist elementary matrices E, E 2,..., E k such that Each of the E i s is invertible and E i A = E k E k E B is also an elementary matrix (Theorem.4.). Thus B = E E 2 E A and hence B is row equivalent to A. 24. (a) If A is row equivalent to B, then there exist elementary matrices E, E 2,..., E k such that A = E k E k E B Since B is row equivalent to C, there exist elementary matrices H, H 2,..., H j such that Thus B = H j H j H C A = E k E k E H j H j H C and hence A is row equivalent to C. (b) If A and B are nonsingular n n matrices, then A and B are row equivalent to I. Since A is row equivalent to I and I is row equivalent to B, it follows from part (a) that A is row equivalent to B. 25. If U is any row echelon form of A, then A can be reduced to U using row operations, so A is row equivalent to U. If B is row equivalent to A, then it follows from the result in Exercise 24(a) that B is row equivalent to U. 26. If B is row equivalent to A, then there exist elementary matrices E, E 2,..., E k such that B = E k E k E A Let M = E k E k E. The matrix M is nonsingular since each of the E i s is nonsingular. Conversely, suppose there exists a nonsingular matrix M such that B = MA. Since M is nonsingular, it is row equivalent to I. Thus, there exist elementary matrices E, E 2,..., E k such that M = E k E k E I It follows that Therefore, B is row equivalent to A. B = MA = E k E k E A k

22 6 Chapter Matrices and Systems of Equations 27. If A is nonsingular, then A is row equivalent to I. If B is row equivalent to A, then using the result from Exercise 24(a), we can conclude that B is row equivalent to I. Therefore, B must be nonsingular. So it is not possible for B to be singular and also be row equivalent to a nonsingular matrix. 28. (a) The system V c = y is given by x x 2 x n x 2 x 2 2 x n 2.. x n+ x 2 n+ x n n+ Comparing the ith row of each side, we have c c 2.. c n+ = y y 2. y n+ c + c 2 x i + + c n+ x n i = y i Thus p(x i ) = y i i =, 2,..., n + (b) If x, x 2,..., x n+ are distinct and V c = 0, then we can apply part (a) with y = 0. Thus if p(x) = c + c 2 x + + c n+ x n, then p(x i ) = 0 i =, 2,..., n + The polynomial p(x) has n + roots. Since the degree of p(x) is less than n +, p(x) must be the zero polynomial. Hence c = c 2 = = c n+ = 0 Since the system V c = 0 has only the trivial solution, the matrix V must be nonsingular. 29. True. If A is row equivalent to I, then A is nonsingular, so if AB = AC, then we can multiply both sides of this equation by A. A AB = A AC B = C 30. True. If E and F are elementary matrices, then they are both nonsingular and the product of two nonsingular matrices is a nonsingular matrix. Indeed, G = F E. 3. True. If a + a 2 = a 3 + 2a 4, then a + a 2 a 3 2a 4 = 0 If we let x = (,,, 2) T, then x is a solution to Ax = 0. Since x 0 the matrix A must be singular. 32. False. Let I be the 2 2 identity matrix and let A = I, B = I, and C = Since B and C are nonsingular, they are both row equivalent to A; however, B + C = is singular, so it cannot be row equivalent to A.

23 Section 6 Partitioned Matrices 7 6 PARTITIONED MATRICES 2. B = A T A = 5. (a) a T a T 2.. a T n 2 2 (c) Let (a, a 2,..., a n ) = a T a a T a 2 a T a n a T 2 a a T 2 a 2 a T 2 a n.. a T n a a T n a 2 a T n a n + ( 2 3) = A = A 2 = (a) A 2 = (0 0) A 22 = ( 0) The block multiplication is performed as follows: A A 2 AT A T 2 A 2 A 22 A T 2 A T = A A T + A 2 A T 2 A A T 2 + A 2 A T A 2 A T + A 22 A T 2 A 2 A T 2 + A 22 A T = XY T = x y T + x 2 y T 2 + x 3 y T 3 = = (b) Since y i x T i = (x i y T i )T for j =, 2, 3, the outer product expansion of Y X T is just the transpose of the outer product expansion of XY T. Thus Y X T = y x T + y 2 x T 2 + y 3 x T 3 = It is possible to perform both block multiplications. To see this, suppose A is a k r matrix, A 2 is a k (n r) matrix, A 2 is an (m k) r matrix and A 22 is (m k) (n r). It is possible to perform the block multiplication of AA T since the matrix multiplications A A T, A A T 2, A 2 A T 2, A 2 A T 22, A 2 A T, A 2 A T 2, A 22 A T 2, A 22 A T 22 are all possible. It is possible to perform the block multiplication of A T A since the matrix multiplications A T A, A T A 2, A T 2A 2, A T 2A, A T 2A 2, A T 22A 2, A T 22A 22 are all possible. 8. AX = A(x, x 2,..., x r ) = (Ax, Ax 2,..., Ax r ) B = (b, b 2,..., b r ) AX = B if and only if the column vectors of AX and B are equal Ax j = b j j =,..., r 9. (a) Since D is a diagonal matrix, its jth column will have d jj in the jth row and the other entries will all be 0. Thus d j = d jj e j for j =,..., n.

24 8 Chapter Matrices and Systems of Equations (b) AD = A(d e, d 22 e 2,..., d nn e n ) = (d Ae, d 22 Ae 2,..., d nn Ae n ) = (d a, d 22 a 2,..., d nn a n ) 0. (a) (b) If we let X = UΣ, then and it follows that. If then Let UΣ = U U 2 Σ = U O Σ + U 2 O = U Σ X = U Σ = (σ u, σ 2 u 2,..., σ n u n ) A = UΣV T = XV T = σ u v T + σ 2 u 2 v T σ n u n v T n A C O A 22 A A 2 I = O A 22 O B = A A 2 + CA 22 = O C = A A 2A 22 A A A 2A 22 O A 22 A A 2 + CA 22 Since AB = BA = I, it follows that B = A. 2. Let 0 denote the zero vector in R n. If A is singular, then there exists a vector x 0 such that Ax = 0. If we set then Mx = A O x = x 0 O x = Ax + O0 = 0 B 0 Ox + B0 0 By Theorem.5.2, M must be singular. Similarly, if B is singular, then there exists a vector x 2 0 such that Bx 2 = 0. So if we set x = 0 x 2 I 5. then x is a nonzero vector and Mx is equal to the zero vector. A = O I, A 2 = I B, A 3 = B I B B I I and hence A + A 2 + A 3 = I + B 2I + B 2I + B I + B I 2B

25 Section 6 Partitioned Matrices 9 6. The block form of S is given by It follows that S MS = I O = I O = O B S = I O A AB I B A AB I B O BA 7. The block multiplication of the two factors yields I O A A 2 = B I O C A I O I O O ABA BA A I A A 2 BA BA 2 + C If we equate this matrix with the block form of A and solve for B and C, we get To check that this works note that and hence I O B I B = A 2 A and C = A 22 A 2 A A 2 BA = A 2 A A = A 2 BA 2 + C = A 2 A A 2 + A 22 A 2 A A 2 = A 22 A A 2 O C = A A 2 A 2 A In order for the block multiplication to work, we must have XB = S and Y M = T = A Since both B and M are nonsingular, we can satisfy these conditions by choosing X = SB and Y = T M. 9. (a) (b) b b 2... BC = b n (c) = Ax = (a, a 2,..., a n ) (c) It follows from parts (a) and (b) that b c b 2 c = cb. b n c x x 2.. x n = a (x ) + a 2 (x 2 ) + + a n (x n ) Ax = a (x ) + a 2 (x 2 ) + + a n (x n ) = x a + x 2 a x n a n

26 20 Chapter Matrices and Systems of Equations 20. If Ax = 0 for all x R n, then a j = Ae j = 0 for j =,..., n and hence A must be the zero matrix. 2. If Bx = Cx for all x R n then 22. (a) It follows from Exercise 20 that (b) If then A 0 A c T A (B C)x = 0 for all x R n c T I A a 0 T c T A a + β B C = O a β B = C x x n+ x x n+ = A 0 c T A = y = A a and z = A b ( c T y + β)x n+ = c T z + b n+ b b n+ A b c T A b + b n+ and x n+ = ct z + b n+ c T y + β x + x n+ A a = A b (β c T y 0) x = A b x n+ A a = z x n+ y MATLAB EXERCISES. In parts (a), (b), (c) it should turn out that A = A4 and A2 = A3. In part (d) A = A3 and A2 = A4. Exact equality will not occur in parts (c) and (d) because of roundoff error. 2. The solution x obtained using the \ operation will be more accurate and yield the smaller residual vector. The computation of x is also more efficient since the solution is computed using Gaussian elimination with partial pivoting and this involves less arithmetic than computing the inverse matrix and multiplying it times b. 3. (a) Since Ax = 0 and x 0, it follows from Theorem.5.2 that A is singular. (b) The columns of B are all multiples of x. Indeed, and hence (c) If D = B + C, then B = (x, 2x, 3x, 4x, 5x, 6x) AB = (Ax, 2Ax, 3Ax, 4Ax, 5Ax, 6Ax) = O AD = AB + AC = O + AC = AC 4. By construction, B is upper triangular whose diagonal entries are all equal to. Thus B is row equivalent to I and hence B is nonsingular. If one changes B by setting b 0, = /256 and computes Bx, the result is the zero vector. Since x 0, the matrix B must be singular.

27 MATLAB Exercises 2 5. (a) Since A is nonsingular, its reduced row echelon form is I. If E,..., E k are elementary matrices such that E k E A = I, then these same matrices can be used to transform (A b) to its reduced row echelon form U. It follows then that U = E k E (A b) = A (A b) = (I A b) Thus, the last column of U should be equal to the solution x of the system Ax = b. (b) After the third column of A is changed, the new matrix A is now singular. Examining the last row of the reduced row echelon form of the augmented matrix (A b), we see that the system is inconsistent. (c) The system Ax = c is consistent since y is a solution. There is a free variable x 3, so the system will have infinitely many solutions. (f) The vector v is a solution since Av = A(w + 3z) = Aw + 3Az = c For this solution, the free variable x 3 = v 3 = 3. To determine the general solution just set x = w + tz. This will give the solution corresponding to x 3 = t for any real number t. 6. (c) There will be no walks of even length from V i to V j whenever i + j is odd. (d) There will be no walks of length k from V i to V j whenever i + j + k is odd. (e) The conjecture is still valid for the graph containing the additional edges. (f) If the edge {V 6, V 8 } is included, then the conjecture is no longer valid. There is now a walk of length from V 6 to V 8 and i + j + k = is odd. 8. The change in part (b) should not have a significant effect on the survival potential for the turtles. The change in part (c) will effect the (2, 2) and (3, 2) of the Leslie matrix. The new values for these entries will be l 22 = and l 32 = With these values, the Leslie population model should predict that the survival period will double but the turtles will still eventually die out. 9. (b) x = c V x2. 0. (b) A 2k = I kb kb I This can be proved using mathematical induction. In the case k = A 2 = O I O I = I B I B I B B I If the result holds for k = m then A 2m+2 A 2m = I mb mb I = A 2 A 2m = I B I mb B I mb I I (m + )B = (m + )B I It follows by mathematical induction that the result holds for all positive integers k.

28 22 Chapter Matrices and Systems of Equations (b) A 2k+ = AA 2k = O I I I B kb kb = kb I I I (k + )B. (a) By construction, the entries of A were rounded to the nearest integer. The matrix B = A T A must also have integer entries and it is symmetric since B T = (A T A) T = A T (A T ) T = A T A = B (b) where It follows that Therefore LDL T = I O B O I E I O F O = B B E T EB E T + F EB ET E = B 2 B and F = B 22 B 2 B B 2 B E T EB I = B (B )T B2 T = B B B 2 = B 2 = B 2 B B = B 2 EB E T + F = B 2 E T + B 22 B 2 B B 2 = B 2 B B 2 + B 22 B 2 B B 2 = B 22 LDL T = B CHAPTER TEST A. The statement is false. If the row echelon form has free variables and the linear system is consistent, then there will be infinitely many solutions. However, it is possible to have an inconsistent system whose coefficient matrix will reduce to an echelon form with free variables. For example, if A = b = 0 0 then A involves one free variable, but the system Ax = b is inconsistent. 2. The statement is true since the zero vector will always be a solution. 3. The statement is true. A matrix A is nonsingular if and only if it is row equivalent to the I (the identity matrix). A will be row equivalent to I if and only if its reduced row echelon form is I. 4. The statement is true. If A is nonsingular, then A is row equivalent to I. So there exist elementary matrices E, E 2,..., E k, such that A = E k E k E I = E k E k E 5. The statement is false. For example, if A = I and B = I, the matrices A and B are both nonsingular, but A + B = O is singular.

29 Chapter Test A The statement is false. For example, if A is any matrix of the form A = cos θ sin θ sin θ cos θ Then A = A. 7. The statement is false. (A B) 2 = A 2 BA AB + B 2 A 2 2AB + B 2 since in general BA AB. For example, if A = and B = then however, (A B) 2 = 0 2 = 0 2 A 2 2AB + B 2 = = The statement is false. If A is nonsingular and AB = AC, then we can multiply both sides of the equation by A and conclude that B = C. However, if A is singular, then it is possible to have AB = AC and B C. For example, if A =, B =, C = then AB = = AC = 2 2 = The statement is false. In general, AB and BA are usually not equal, so it is possible for AB = O and BA to be a nonzero matrix. For example, if A = and B = then AB = 0 0 and BA = The statement is true. If x = (, 2, ) T, then x 0 and Ax = 0, so A must be singular.. The statement is true. If b = a + a 3 and x = (, 0, ) T, then Ax = x a + x 2 a 2 + x 3 a 3 = a + 0a 2 + a 3 = b So x is a solution to Ax = b. 2. The statement is true. If b = a + a 2 + a 3, then x = (,, ) T is a solution to Ax = b, since Ax = x a + x 2 a 2 + x 3 a 3 = a + a 2 + a 3 = b If a 2 = a 3, then we can also express b as a linear combination b = a + 0a 2 + 2a 3

30 24 Chapter Matrices and Systems of Equations Thus y = (, 0, 2) T is also a solution to the system. However, if there is more than one solution, then the reduced row echelon form of A must involve a free variable. A consistent system with a free variable must have infinitely many solutions. 3. The statement is true. An elementary matrix E of type I or type II is symmetric. So in either case we have E T = E is elementary. If E is an elementary matrix of type III formed from the identity matrix by adding a nonzero multiple c of row k to row j, then E T will be the elementary matrix of type III formed from the identity matrix by adding c times row j to row k. 4. The statement is false. An elementary matrix is a matrix that is constructed by performing exactly one elementary row operation on the identity matrix. The product of two elementary matrices will be a matrix formed by performing two elementary row operations on the identity matrix. For example, E = 2 0 and E 2 = are elementary matrices, however; 0 0 E E 2 = is not an elementary matrix. 5. The statement is true. The row vectors of A are x y T, x 2 y T,..., x n y T. Note, all of the row vectors are multiples of y T. Since x and y are nonzero vectors, at least one of these row vectors must be nonzero. However, if any nonzero row is picked as a pivot row, then since all of the other rows are multiples of the pivot row, they will all be eliminated in the first step of the reduction process. The resulting row echelon form will have exactly one nonzero row. CHAPTER TEST B The free variables are x 2 and x 4. If we set x 2 = a and x 4 = b, then x = 4 + a + 7b and x 3 = 3b and hence the solution set consists of all vectors of the form 4 + a + 7b a x = 3b b 2. (a) A linear equation in 3 unknowns corresponds to a plane in 3-space. (b) Given 2 equations in 3 unknowns, each equation corresponds to a plane. If one equation is a multiple of the other, then the equations represent the same plane and any point on the that plane will be a solution to the system. If the two planes are distinct, then they are either parallel or they intersect in a line. If they are parallel they do not intersect, so

LINEAR ALGEBRA WITH APPLICATIONS

LINEAR ALGEBRA WITH APPLICATIONS SEVENTH EDITION LINEAR ALGEBRA WITH APPLICATIONS Instructor s Solutions Manual Steven J. Leon PREFACE This solutions manual is designed to accompany the seventh edition of Linear Algebra with Applications

More information

MAT 2037 LINEAR ALGEBRA I web:

MAT 2037 LINEAR ALGEBRA I web: MAT 237 LINEAR ALGEBRA I 2625 Dokuz Eylül University, Faculty of Science, Department of Mathematics web: Instructor: Engin Mermut http://kisideuedutr/enginmermut/ HOMEWORK 2 MATRIX ALGEBRA Textbook: Linear

More information

Chapter 1 Matrices and Systems of Equations

Chapter 1 Matrices and Systems of Equations Chapter 1 Matrices and Systems of Equations System of Linear Equations 1. A linear equation in n unknowns is an equation of the form n i=1 a i x i = b where a 1,..., a n, b R and x 1,..., x n are variables.

More information

Systems of Linear Equations and Matrices

Systems of Linear Equations and Matrices Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first

More information

Systems of Linear Equations and Matrices

Systems of Linear Equations and Matrices Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first

More information

Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations

Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2

More information

Fundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved

Fundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved Fundamentals of Linear Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 PREFACE Linear algebra has evolved as a branch of mathematics with wide range of applications to the natural

More information

Fundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved

Fundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved Fundamentals of Linear Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved October 9, 200 2 PREFACE Linear algebra has evolved as a branch of mathematics with wide range of applications

More information

Linear Systems and Matrices

Linear Systems and Matrices Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.......

More information

Math113: Linear Algebra. Beifang Chen

Math113: Linear Algebra. Beifang Chen Math3: Linear Algebra Beifang Chen Spring 26 Contents Systems of Linear Equations 3 Systems of Linear Equations 3 Linear Systems 3 2 Geometric Interpretation 3 3 Matrices of Linear Systems 4 4 Elementary

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

Linear Algebra and Matrix Inversion

Linear Algebra and Matrix Inversion Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much

More information

Lecture Notes in Linear Algebra

Lecture Notes in Linear Algebra Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................

More information

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1

More information

Extra Problems: Chapter 1

Extra Problems: Chapter 1 MA131 (Section 750002): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 1 Extra Problems: Chapter 1 1. In each of the following answer true if the statement is always true and false otherwise in the space

More information

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column

More information

Linear Algebra Highlights

Linear Algebra Highlights Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to

More information

GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511)

GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) D. ARAPURA Gaussian elimination is the go to method for all basic linear classes including this one. We go summarize the main ideas. 1.

More information

1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det

1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix

More information

LINEAR SYSTEMS AND MATRICES

LINEAR SYSTEMS AND MATRICES CHAPTER 3 LINEAR SYSTEMS AND MATRICES SECTION 3. INTRODUCTION TO LINEAR SYSTEMS This initial section takes account of the fact that some students remember only hazily the method of elimination for and

More information

Linear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.

Linear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0. Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the

More information

Linear Algebra: Lecture notes from Kolman and Hill 9th edition.

Linear Algebra: Lecture notes from Kolman and Hill 9th edition. Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices

More information

Linear Algebra March 16, 2019

Linear Algebra March 16, 2019 Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented

More information

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the

More information

College Algebra. Third Edition. Concepts Through Functions. Michael Sullivan. Michael Sullivan, III. Chicago State University. Joliet Junior College

College Algebra. Third Edition. Concepts Through Functions. Michael Sullivan. Michael Sullivan, III. Chicago State University. Joliet Junior College College Algebra Concepts Through Functions Third Edition Michael Sullivan Chicago State University Michael Sullivan, III Joliet Junior College PEARSON Boston Columbus Indianapolis New York San Francisco

More information

ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,

More information

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017 Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...

More information

Section 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns.

Section 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. Section 9.2: Matrices Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. That is, a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn A

More information

Chapter 1: Systems of Linear Equations and Matrices

Chapter 1: Systems of Linear Equations and Matrices : Systems of Linear Equations and Matrices Multiple Choice Questions. Which of the following equations is linear? (A) x + 3x 3 + 4x 4 3 = 5 (B) 3x x + x 3 = 5 (C) 5x + 5 x x 3 = x + cos (x ) + 4x 3 = 7.

More information

Matrix & Linear Algebra

Matrix & Linear Algebra Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A

More information

Essentials of College Algebra

Essentials of College Algebra Essentials of College Algebra For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects and requirements, equipping students

More information

a s 1.3 Matrix Multiplication. Know how to multiply two matrices and be able to write down the formula

a s 1.3 Matrix Multiplication. Know how to multiply two matrices and be able to write down the formula Syllabus for Math 308, Paul Smith Book: Kolman-Hill Chapter 1. Linear Equations and Matrices 1.1 Systems of Linear Equations Definition of a linear equation and a solution to a linear equations. Meaning

More information

Linear Algebra Homework and Study Guide

Linear Algebra Homework and Study Guide Linear Algebra Homework and Study Guide Phil R. Smith, Ph.D. February 28, 20 Homework Problem Sets Organized by Learning Outcomes Test I: Systems of Linear Equations; Matrices Lesson. Give examples of

More information

Matrix operations Linear Algebra with Computer Science Application

Matrix operations Linear Algebra with Computer Science Application Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the

More information

MTH Linear Algebra. Study Guide. Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education

MTH Linear Algebra. Study Guide. Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education MTH 3 Linear Algebra Study Guide Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education June 3, ii Contents Table of Contents iii Matrix Algebra. Real Life

More information

MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION

MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether

More information

MATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018

MATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018 Homework #1 Assigned: August 20, 2018 Review the following subjects involving systems of equations and matrices from Calculus II. Linear systems of equations Converting systems to matrix form Pivot entry

More information

Math 313 Chapter 1 Review

Math 313 Chapter 1 Review Math 313 Chapter 1 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 1.1 Introduction to Systems of Linear Equations 2 2 1.2 Gaussian Elimination 3 3 1.3 Matrices and Matrix Operations

More information

a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.

a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real

More information

Equality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.

Equality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same. Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read

More information

Linear Algebra M1 - FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8.

Linear Algebra M1 - FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8. Linear Algebra M1 - FIB Contents: 5 Matrices, systems of linear equations and determinants 6 Vector space 7 Linear maps 8 Diagonalization Anna de Mier Montserrat Maureso Dept Matemàtica Aplicada II Translation:

More information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

Lecture Summaries for Linear Algebra M51A

Lecture Summaries for Linear Algebra M51A These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture

More information

Extra Problems for Math 2050 Linear Algebra I

Extra Problems for Math 2050 Linear Algebra I Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as

More information

MAT Linear Algebra Collection of sample exams

MAT Linear Algebra Collection of sample exams MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system

More information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

Matrix Algebra. Matrix Algebra. Chapter 8 - S&B

Matrix Algebra. Matrix Algebra. Chapter 8 - S&B Chapter 8 - S&B Algebraic operations Matrix: The size of a matrix is indicated by the number of its rows and the number of its columns. A matrix with k rows and n columns is called a k n matrix. The number

More information

APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF

APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF ELEMENTARY LINEAR ALGEBRA WORKBOOK/FOR USE WITH RON LARSON S TEXTBOOK ELEMENTARY LINEAR ALGEBRA CREATED BY SHANNON MARTIN MYERS APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF When you are done

More information

HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)

HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe

More information

Fundamentals of Engineering Analysis (650163)

Fundamentals of Engineering Analysis (650163) Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is

More information

Review for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.

Review for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for

More information

Numerical Linear Algebra Homework Assignment - Week 2

Numerical Linear Algebra Homework Assignment - Week 2 Numerical Linear Algebra Homework Assignment - Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.

More information

EXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1)

EXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1) EXERCISE SET 5. 6. The pair (, 2) is in the set but the pair ( )(, 2) = (, 2) is not because the first component is negative; hence Axiom 6 fails. Axiom 5 also fails. 8. Axioms, 2, 3, 6, 9, and are easily

More information

Section 9.2: Matrices.. a m1 a m2 a mn

Section 9.2: Matrices.. a m1 a m2 a mn Section 9.2: Matrices Definition: A matrix is a rectangular array of numbers: a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn In general, a ij denotes the (i, j) entry of A. That is, the entry in

More information

MATH10212 Linear Algebra B Homework Week 5

MATH10212 Linear Algebra B Homework Week 5 MATH Linear Algebra B Homework Week 5 Students are strongly advised to acquire a copy of the Textbook: D C Lay Linear Algebra its Applications Pearson 6 (or other editions) Normally homework assignments

More information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

More information

Chapter 2: Matrix Algebra

Chapter 2: Matrix Algebra Chapter 2: Matrix Algebra (Last Updated: October 12, 2016) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). Write A = 1. Matrix operations [a 1 a n. Then entry

More information

Chapter 2: Matrices and Linear Systems

Chapter 2: Matrices and Linear Systems Chapter 2: Matrices and Linear Systems Paul Pearson Outline Matrices Linear systems Row operations Inverses Determinants Matrices Definition An m n matrix A = (a ij ) is a rectangular array of real numbers

More information

Elementary Linear Algebra with Applications Bernard Kolman David Hill Ninth Edition

Elementary Linear Algebra with Applications Bernard Kolman David Hill Ninth Edition Elementary Linear Algebra with Applications Bernard Kolman David Hill Ninth Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM JE England and Associated Companies throughout the world Visit

More information

Matrix Algebra for Engineers Jeffrey R. Chasnov

Matrix Algebra for Engineers Jeffrey R. Chasnov Matrix Algebra for Engineers Jeffrey R. Chasnov The Hong Kong University of Science and Technology The Hong Kong University of Science and Technology Department of Mathematics Clear Water Bay, Kowloon

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v 1,, v p } in n is said to be linearly independent if the vector equation x x x

More information

Introduction to Matrices and Linear Systems Ch. 3

Introduction to Matrices and Linear Systems Ch. 3 Introduction to Matrices and Linear Systems Ch. 3 Doreen De Leon Department of Mathematics, California State University, Fresno June, 5 Basic Matrix Concepts and Operations Section 3.4. Basic Matrix Concepts

More information

Linear Algebra Summary. Based on Linear Algebra and its applications by David C. Lay

Linear Algebra Summary. Based on Linear Algebra and its applications by David C. Lay Linear Algebra Summary Based on Linear Algebra and its applications by David C. Lay Preface The goal of this summary is to offer a complete overview of all theorems and definitions introduced in the chapters

More information

OHSx XM511 Linear Algebra: Solutions to Online True/False Exercises

OHSx XM511 Linear Algebra: Solutions to Online True/False Exercises This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)

More information

A matrix over a field F is a rectangular array of elements from F. The symbol

A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted

More information

Solving Linear Systems Using Gaussian Elimination

Solving Linear Systems Using Gaussian Elimination Solving Linear Systems Using Gaussian Elimination DEFINITION: A linear equation in the variables x 1,..., x n is an equation that can be written in the form a 1 x 1 +...+a n x n = b, where a 1,...,a n

More information

Chapter 7. Linear Algebra: Matrices, Vectors,

Chapter 7. Linear Algebra: Matrices, Vectors, Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.

More information

Knowledge Discovery and Data Mining 1 (VO) ( )

Knowledge Discovery and Data Mining 1 (VO) ( ) Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory

More information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

More information

MATH 2030: MATRICES. Example 0.2. Q:Define A 1 =, A. 3 4 A: We wish to find c 1, c 2, and c 3 such that. c 1 + c c

MATH 2030: MATRICES. Example 0.2. Q:Define A 1 =, A. 3 4 A: We wish to find c 1, c 2, and c 3 such that. c 1 + c c MATH 2030: MATRICES Matrix Algebra As with vectors, we may use the algebra of matrices to simplify calculations. However, matrices have operations that vectors do not possess, and so it will be of interest

More information

Finite Mathematics Chapter 2. where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero.

Finite Mathematics Chapter 2. where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero. Finite Mathematics Chapter 2 Section 2.1 Systems of Linear Equations: An Introduction Systems of Equations Recall that a system of two linear equations in two variables may be written in the general form

More information

Notes on Mathematics

Notes on Mathematics Notes on Mathematics - 12 1 Peeyush Chandra, A. K. Lal, V. Raghavendra, G. Santhanam 1 Supported by a grant from MHRD 2 Contents I Linear Algebra 7 1 Matrices 9 1.1 Definition of a Matrix......................................

More information

Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes.

Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes. Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes Matrices and linear equations A matrix is an m-by-n array of numbers A = a 11 a 12 a 13 a 1n a 21 a 22 a 23 a

More information

A FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Matrix Arithmetic

A FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Matrix Arithmetic A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Matrix Arithmetic Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)

More information

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column

More information

The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver I.N.

The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver I.N. Math 410 Homework Problems In the following pages you will find all of the homework problems for the semester. Homework should be written out neatly and stapled and turned in at the beginning of class

More information

2. Every linear system with the same number of equations as unknowns has a unique solution.

2. Every linear system with the same number of equations as unknowns has a unique solution. 1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations

More information

Math Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT

Math Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear

More information

MATH 2331 Linear Algebra. Section 1.1 Systems of Linear Equations. Finding the solution to a set of two equations in two variables: Example 1: Solve:

MATH 2331 Linear Algebra. Section 1.1 Systems of Linear Equations. Finding the solution to a set of two equations in two variables: Example 1: Solve: MATH 2331 Linear Algebra Section 1.1 Systems of Linear Equations Finding the solution to a set of two equations in two variables: Example 1: Solve: x x = 3 1 2 2x + 4x = 12 1 2 Geometric meaning: Do these

More information

a11 a A = : a 21 a 22

a11 a A = : a 21 a 22 Matrices The study of linear systems is facilitated by introducing matrices. Matrix theory provides a convenient language and notation to express many of the ideas concisely, and complicated formulas are

More information

Offline Exercises for Linear Algebra XM511 Lectures 1 12

Offline Exercises for Linear Algebra XM511 Lectures 1 12 This document lists the offline exercises for Lectures 1 12 of XM511, which correspond to Chapter 1 of the textbook. These exercises should be be done in the traditional paper and pencil format. The section

More information

MATH 369 Linear Algebra

MATH 369 Linear Algebra Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine

More information

Jim Lambers MAT 610 Summer Session Lecture 1 Notes

Jim Lambers MAT 610 Summer Session Lecture 1 Notes Jim Lambers MAT 60 Summer Session 2009-0 Lecture Notes Introduction This course is about numerical linear algebra, which is the study of the approximate solution of fundamental problems from linear algebra

More information

Online Exercises for Linear Algebra XM511

Online Exercises for Linear Algebra XM511 This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2

More information

MATH 240 Spring, Chapter 1: Linear Equations and Matrices

MATH 240 Spring, Chapter 1: Linear Equations and Matrices MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear

More information

Linear Algebra. Linear Equations and Matrices. Copyright 2005, W.R. Winfrey

Linear Algebra. Linear Equations and Matrices. Copyright 2005, W.R. Winfrey Copyright 2005, W.R. Winfrey Topics Preliminaries Systems of Linear Equations Matrices Algebraic Properties of Matrix Operations Special Types of Matrices and Partitioned Matrices Matrix Transformations

More information

1 Determinants. 1.1 Determinant

1 Determinants. 1.1 Determinant 1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to

More information

7 Matrix Operations. 7.0 Matrix Multiplication + 3 = 3 = 4

7 Matrix Operations. 7.0 Matrix Multiplication + 3 = 3 = 4 7 Matrix Operations Copyright 017, Gregory G. Smith 9 October 017 The product of two matrices is a sophisticated operations with a wide range of applications. In this chapter, we defined this binary operation,

More information

MATH2210 Notebook 2 Spring 2018

MATH2210 Notebook 2 Spring 2018 MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................

More information

Linear Algebra. Solving Linear Systems. Copyright 2005, W.R. Winfrey

Linear Algebra. Solving Linear Systems. Copyright 2005, W.R. Winfrey Copyright 2005, W.R. Winfrey Topics Preliminaries Echelon Form of a Matrix Elementary Matrices; Finding A -1 Equivalent Matrices LU-Factorization Topics Preliminaries Echelon Form of a Matrix Elementary

More information

Math Linear Algebra Final Exam Review Sheet

Math Linear Algebra Final Exam Review Sheet Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

More information

2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian

2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian FE661 - Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian

More information

Exercise Set Suppose that A, B, C, D, and E are matrices with the following sizes: A B C D E

Exercise Set Suppose that A, B, C, D, and E are matrices with the following sizes: A B C D E Determine the size of a given matrix. Identify the row vectors and column vectors of a given matrix. Perform the arithmetic operations of matrix addition, subtraction, scalar multiplication, and multiplication.

More information

Math 54 HW 4 solutions

Math 54 HW 4 solutions Math 54 HW 4 solutions 2.2. Section 2.2 (a) False: Recall that performing a series of elementary row operations A is equivalent to multiplying A by a series of elementary matrices. Suppose that E,...,

More information

Boundary. DIFFERENTIAL EQUATIONS with Fourier Series and. Value Problems APPLIED PARTIAL. Fifth Edition. Richard Haberman PEARSON

Boundary. DIFFERENTIAL EQUATIONS with Fourier Series and. Value Problems APPLIED PARTIAL. Fifth Edition. Richard Haberman PEARSON APPLIED PARTIAL DIFFERENTIAL EQUATIONS with Fourier Series and Boundary Value Problems Fifth Edition Richard Haberman Southern Methodist University PEARSON Boston Columbus Indianapolis New York San Francisco

More information

Math Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88

Math Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88 Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant

More information

ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Corrected Version, 7th April 013 Comments to the author at keithmatt@gmail.com Chapter 1 LINEAR EQUATIONS 1.1

More information

Elementary Row Operations on Matrices

Elementary Row Operations on Matrices King Saud University September 17, 018 Table of contents 1 Definition A real matrix is a rectangular array whose entries are real numbers. These numbers are organized on rows and columns. An m n matrix

More information

Problem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show

Problem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show MTH 0: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology - Kanpur Problem Set Problems marked (T) are for discussions in Tutorial sessions (T) If A is an m n matrix,

More information

MATH 2360 REVIEW PROBLEMS

MATH 2360 REVIEW PROBLEMS MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) 1 2 2 1 (a) 0 1 1 2 ( ) 0 1 2 (b) 0 3 1 4 3 4 5 2 5 (c) 0 3 ) 1 4 ( 1

More information