Linear Algebra and Its Applications 4th Edition

Exam Name___________________________________ SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.Solve the problem. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.Solve the problem. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.Find the matrix product AB for the partitioned matrices. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.Find the matrix product AB for the partitioned matrices. MULTIPLE CHOICE. Choose the one alternative

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Compute the determinant of the matrix by cofactor expansion. 1) 2-2 6 4 2 2 7 1 6-614 14 -2 2 -6 1 1) A) 80 B) -40 C) -160 D) -80 Answer:CExplanation:A)B)C)D) Answer:CExplanation:A)B)C)D) Answer: C Explanation: A) B) C) D) 2) 4 1 5 5 4 3 4 1 4 2) A) -11 B) 11 C) 213 D) -35 Answer:AExplanation:A)B)C)D) Answer:AExplanation:A)B)C)D) Answer: A Explanation: A) B) C) D) 3) -1 2 -1 4 3 -2 -3-4-1 3) A) -38 B) 38 C) 14 D) 24 Answer:BExplanation:A)B)C)D) Answer:BExplanation:A)B)C)D) Answer: B Explanation: A) B) C) D) 4) 8 0 0 7-4 0 -4 2 2 4) A) -64 B) 50 C) 64 D) -78 Answer:AExplanation:A)B)C)D) Answer:AExplanation:A)B)C)D) Answer: A Explanation: A) B) C) D) 1 5) 4 1 2 -4-1-5 8 5

Exam Name___________________________________ SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 1) A mathematician has found 4 solutions to a homogeneous system of 36 equations in 39 variables. The 4 solutions are linearly independent and all other solutions can be constructed by adding together appropriate multiples of

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. For the given matrix A, find a basis for the corresponding eigenspace for the given eigenvalue. 1) A =1- 4 - 4 - 4 1 4 4- 4 - 7 , = -3 1) A) 0 1 -1 B) 1 0 1, 0 1 -1 C) 1 0 -1, 0 1 1 D) 1 0 -1 Answer:BExplanation:A)B)C)D) Answer:BExplanation:A)B)C)D) Answer: B Explanation: A) B) C) D) Find the eigenvalues of A, and

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find a QR factorization of the matrix A. 1) A = 0 1 1 1 1 0 -1-1 1 1 -1 1 1) A) Q = 0 3 14 1 2 2 -1-2 9 1 -4 7 , R = 330 0 1311 33 0 0-3 33 30 330 B) Q =

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find a singular value decomposition of the matrix A. 1) A =-9 0 0 5 1) A) A = 1 0 0 1 -9 0 0 51 0 0 1 B) A =-1 0 0 1 9 0 0 5 1 0 0 1 C) A =-9 0

Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it. 1) 1 1, 3 5, 0 6, p= 11 2 -1 2 1) A) 2, 5 2, -7 2 B) 1, 3 2, -3 2 C) (-9, 3, 7) D) 3 2, -3 2, 1 Answer:BExplanation:A)B)C)D) Answer:BExplanation:A)B)C)D) Answer: B Explanation: A) B) C) D) Provide an appropriate response. 2) Suppose {v1, v2, v3} is

Exam Name___________________________________ SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) A set S in Rn is convex if, for each p and q in S, the line segment between p and q lies in S. [This line segment is the set of points

INSTRUCTORS TI CALCULATOR MANUAL MICHAEL MILLER Corban University LUZ DEALBA Drake University LINEAR ALGEBRA AND ITS APPLICATIONS FOURTH EDITION David C. Lay University of Maryland

TI Project: Cryptography 1 Time saving tip: For repeated row operations it is convenient to store the partial result into a named matrix in memory. For example mRowAdd(7,aa,1,3)!aa replaces A with a new matrix obtained from A by replacing the third row with itself plus 7 times the first

TI Project: Linear Transformations 1 TI Project: Linear Transformations The problems will help you to visualize the action of linear transformations on the plane, and to view matrices as functions acting on vectors. Review Sections 1.8 and 1.9 of the text. Data for this project is in

17 17 17 5. References [1] D. Carlson, C. R. Johnson, D. C. Lay, A. D. Porter, Ann E. Watkins, and William Watkins, eds., "Resources for Teaching Linear Algebra", MAA Notes #42, Mathematical Association of America, Washington, D.C., 1999. [2] D. Carlson, C. R. Johnson, D. C. Lay,

74 CHAPTER 1 Linear Equations in Linear Algebra 74 CHAPTER 1 Linear Equations in Linear Algebra 74 CHAPTER 1 Linear Equations in Linear Algebra Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 7.

1.3 Solutions 21 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 3100 6400 23610 ~ 0 1.000 1.800250 360 1623 0 0 0 The steam plant burned 3.9 tons of anthracite coal and

1.6 Solutions 41 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Move all variables to the left side and combine like terms: .8 .9 .4 0.1 .8 .6 0FMSFMSpp ppp p+=+= .8

Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 1 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 1 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 1 1.1 SOLUTIONS Notes: The key exercises are 7 (or 11 or 12), 1922, and 25.

1.8 Solutions 61 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. which is not the same as T(x) + T(y) = Ax + b + Ay + b which is not the

2.3 Solutions 107 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. ( ) ( ( )) ( ( )) Because is linearByequation (1)()Sr SrT STr TrrS====ux xxu So S preserves scalar multiples. Thus

Copyright 2012 Peason Education, Inc. Publishing as Addison-Wesley. 87 Copyright 2012 Peason Education, Inc. Publishing as Addison-Wesley. 87 Copyright 2012 Pea son Education, Inc. Publishing as Addison-Wesley. 87 r 2.1 SOLUTIONS Notes: The definition here of a matri x calculations. (The dual fact about the rovectors here are usually

2.5 Solutions 127 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 1100 100~0 1 0 2 1 0 ,001 111IL=1100so 2 1 0 .111L= 1100 100~0 1 0 2 1 0 ,001

2.8 Solutions 147 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 5. The vector w is in the subspace generated by v 1 and v 2 if and only if the vector equation x 1 v 1 +

182 CHAPTER 3 Determinants Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 5. The system is equivalent to Ax = b, where 210 301 012 A = and 7 8 3 = b . We compute 12 3() 8 0 1, ()

Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 167 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 167 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 167 3.1 SOLUTIONS Notes: Some exercises in this section provide practice in computing determinants, while others

Copyright 2012 Peason Education, Inc. Publishing as Addison-Wesley. 197 Copyright 2012 Peason Education, Inc. Publishing as Addison-Wesley. 197 Copyright 2012 Pea son Education, Inc. Publishing as Addison-Wesley. 197 r 4.1 SOLUTIONS Notes: This section is designed to avoi axioms on an array of sets. Theorem 1 pset is

4.3 Solutions 217 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 232 44510,020100xxx xxx == + x 232 44510,020100xxx xxx == + x 2 32 4 4 5 10 , 02 01 00 x xx x x x == +

4.8 Solutions 255 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 6876 .35 .5 .35 .35(3) .5(.7) .35(0) 1.4,zyyy=++= ++ = 7987.35 .5 .35 .35(.7) .5(3) .35(.7) 2,zyyy=++= ++ = 81098.35 .5 .35

4.6 Solutions 237 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. A basis for Row A is the pivot rows of B: { } (2,6, 6,6,3,6),(0,3,0,3,3,0),(0,0,0,0,3,0) . To find a basis for Nul A row

5.8 Solutions 345 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Solving this equation for Ax, we find that 11() =+=+ xxx xA Thus (1 )= + / is an eigenvalue of

5.7 Solutions 333 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. (3 ) ( 3 )1233 3322it itiicec e+ + (3 ) ( 3 )1233 3322it itiicec e+ + (3 )

5.3 Solutions 293 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 14. The eigenvalues of A are given to be 2 and 3. For = 2: 00 2 2112, 00 1 AI = general solution is 211,0x

Copyright 2012 Peason Education, Inc. Publishing as Addison-Wesley. 273 Copyright 2012 Peason Education, Inc. Publishing as Addison-Wesley. 273 Copyright 2012 Pea son Education, Inc. Publishing as Addison-Wesley. 273 r 5.1 SOLUTIONS Notes : Exercises 16 reinforce the d e eigenvectors and difference equations, a example and anticipates discussions of d 1. The

5.5 Solutions 313 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 3. 51 . 81 A = The characteristic polynomial is 2 613,+ so the eigenvalues of A are 6365232.2i== For = 3 + 2i: 22 1(3 2

Copyright 2012 Peason Education, Inc. Publishing as Addison-Wesley. 357 Copyright 2012 Peason Education, Inc. Publishing as Addison-Wesley. 357 Copyright 2012 Pea son Education, Inc. Publishing as Addison-Wesley. 357 r 6.1 SOLUTIONS Notes: The first half of this section is c concepts of orthogonality and orthogon a an important general fact, but

6.7 Solutions 391 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley 6.7 SOLUTIONS Notes : The three types of inner products described here (in Examples 1, 2, and 7) are matched by examples in Section

6.4 Solutions 377 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley 1/ 5 0 0 5545,0621/ 5 1/2 1/20041/ 5 1/2 1/21/ 5 1/2 1/2TQRQA=== 1/ 5 0 0 5545,0621/ 5 1/2

Copyright 2012 Peason Education, Inc. Publishing as Addison-Wesley. 405 Copyright 2012 Peason Education, Inc. Publishing as Addison-Wesley. 405 Copyright 2012 Pea son Education, Inc. Publishing as Addison-Wesley. 405 r 7.1 SOLUTIONS Notes: Students can profit by reviewin working on this section. Theorems 1 an d sections that follow. Note that symmetri c text

7.4 Solutions 439 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 1/ 2 1/ 2 25: , 9: 1/ 2 1/ 2 == Thus one choice for V is 1/ 2

7.2 Solutions 425 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 17. [M] The matrix of the quadratic form is 0619/2609/21A= The eigenvalues of A are 17. [M] The matrix of the

Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 453 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 453 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 453 The Geometry of Vector Spaces 8.1 SOLUTIONS Notes . This section introduces a special kinds

8.4 Solutions 467 Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley. h(t) = (1 t)3p0 + 3t(1 t)2p1 + 3t2(1 t)p2 + t 3p3 for 0 t 1 Thus, h(t) is in the convex

INSTRUCTORS MAPLE MANUAL DOUGLAS MEADE University of South Carolina LINEAR ALGEBRA AND ITS APPLICATIONS FOURTH EDITION David C. Lay University of Maryland Reproduced by Pearson Addison-Wesley from electronic files supplied by the author. Copyright 2012, 2006, 1997 Pearson Education, Inc. Publishing as Pearson Addison-Wesley, 75 Arlington Street,

3. Let A = [v1v2. . . vk] be a matrix whose columns are v1,v2, . . . , vk. Explain why the following are logically equivalent:The set of vectors {v1,v2, . . . , vk}is linearly independent.The rank of A is k. following are logically equivalent:The set of vectors {v1,v2, .

COMPUTER PROJECTS 17 Chapter 4: Space Flight and Control Systems This case study studies a mathematical model for engineering control systems. The central idea is that the rank of a matrix determines if a system is controllable. A system of linear equations is solved to determine which inputs into the system yield a desired

Maple Project: Temperature Distributions NameMaple Project: Temperature Distributions NameMaple Project: Temperature Distributions Name Purpose Given the temperature on a regions boundary, determine the steady- state temperature inside. Prerequisites Section 2.5 Maple commands used IdentityMatrix and MatrixInverse from the LinearAlgebra pack- age. Consider a situation in which the temperature is held constant on the surfaces of a

3. The five matrices (A, B, C, D, and E ) can be loaded into a Maple session with the command: cxeigdat( );. Recall that the Maple name for matrix D is DD. The eigenvectors command can be used to obtain all eigenvalues and the corresponding eigenvectors for a matrix. Use the following

Page 3 of 4 MATLAB Project: Partitioned Matrices Page 3 of 4 MATLAB Project: Partitioned Matrices Page 3 of 4 MATLAB Project: Partitioned Matrices Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison

*It is suggested that this project not be assigned until after Section 1.5 or 1.6. Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley MATLAB PROJECTS to

INSTRUCTORS MATLAB MANUAL JEREMY R. CASE Taylor University JANE DAY San Jose State University LINEAR ALGEBRA AND ITS APPLICATIONS FOURTH EDITION David C. Lay University of Maryland Reproduced by Pearson Addison-Wesley from electronic files supplied by the author. Copyright 2012, 2006, 1997 Pearson Education, Inc. Publishing as

Page 2 of 2 MATLAB Project: Lower Triangular Matrices Page 2 of 2 MATLAB Project: Lower Triangular Matrices Page 2 of 2 MATLAB Project: Lower Triangular Matrices Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley Copyright 2012 Pearson Education, Inc.

Page 2 of 3 MATLAB Project: Reduced Echelon Form and ref Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley 2. Let B = 7.65.00 7.02.03.0 21.01.0 . (a) (hand) Calculate the reduced echelon

Page 5 of 5 MATLAB Project: Using Eigenvalues to Study Spotted Owls Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley (c) Prove that the real eigenvalue 1 is greater

Page 4 of 4 MATLAB Project: Homogeneous Coordinates for Computer Graphics Page 4 of 4 MATLAB Project: Homogeneous Coordinates for Computer Graphics Page 4 of 4 MATLAB Project: Homogeneous Coordinates for Computer Graphics Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley Copyright 2012 Pearson Education, Inc. Publishing as Addison