Exercises and Problems in Linear Algebra

1 Exercises and Problems in Linear AlgebraJohn M. ErdmanPortland State UniversityVersion July 13, 2014c 2010 John M. ErdmanE-mail 1. MATRICES AND Linear EQUATIONS1 Chapter 1. SYSTEMS OF Linear Answers to Odd-Numbered Exercises8 Chapter 2. ARITHMETIC OF Answers to Odd-Numbered Exercises14 Chapter 3. ELEMENTARY MATRICES; Answers to Odd-Numbered Exercises23 Chapter 4. VECTOR GEOMETRY Answers to Odd-Numbered Exercises29 Part 2. VECTOR SPACES31 Chapter 5. VECTOR Answers to Odd-Numbered Exercises38 Chapter 6. Answers to Odd-Numbered Exercises45 Chapter 7. Linear Answers to Odd-Numbered Exercises53 Chapter 8.

2 BASIS FOR A VECTOR Answers to Odd-Numbered Exercises58 Part 3. Linear MAPS BETWEEN VECTOR SPACES59 Chapter 9. Answers to Odd-Numbered Exercises70 Chapter 10. Linear MAPS BETWEEN EUCLIDEAN Answers to Odd-Numbered Exercises75 Chapter 11. PROJECTION Answers to Odd-Numbered Exercises80 Part 4. SPECTRAL THEORY OF VECTOR SPACES81 Chapter 12. EIGENVALUES AND Answers to Odd-Numbered Exercises86 Chapter 13. DIAGONALIZATION OF Answers to Odd-Numbered Exercises92 Chapter 14. SPECTRAL THEOREM FOR VECTOR Answers to Odd-Numbered Exercises96 Chapter 15. SOME APPLICATIONS OF THE SPECTRAL Answers to Odd-Numbered Exercises103 Chapter 16.

3 EVERY OPERATOR IS DIAGONALIZABLE PLUS Answers to Odd-Numbered Exercises111 Part 5. THE GEOMETRY OF INNER PRODUCT SPACES113 Chapter 17. COMPLEX Answers to Odd-Numbered Exercises119 Chapter 18. REAL AND COMPLEX INNER PRODUCT Answers to Odd-Numbered Exercises126 Chapter 19. ORTHONORMAL SETS OF Answers to Odd-Numbered Exercises131 Chapter 20. QUADRATIC Answers to Odd-Numbered Exercises137 Chapter 21. Answers to Odd-Numbered Exercises142 Part 6. ADJOINT OPERATORS143 Chapter 22. ADJOINTS AND Answers to Odd-Numbered Exercises148 Chapter 23. THE FOUR FUNDAMENTAL Answers to Odd-Numbered Exercises157 Chapter 24.

4 ORTHOGONAL Answers to Odd-Numbered Exercises164 Chapter 25. LEAST SQUARES Answers to Odd-Numbered Exercises168 Part 7. SPECTRAL THEORY OF INNER PRODUCT SPACES169 Chapter 26. SPECTRAL THEOREM FOR REAL INNER PRODUCT Answers to the Odd-Numbered Exercise175 Chapter 27. SPECTRAL THEOREM FOR COMPLEX INNER PRODUCT Answers to Odd-Numbered Exercises182 Bibliography183 Index185 PREFACEThis collection of Exercises is designed to provide a framework for discussion in a junior levellinear Algebra class such as the one I have conducted fairly regularly at Portland State is no assigned text. Students are free to choose their own sources of information.

5 Stu-dents are encouraged to find books, papers, and web sites whose writing style they find congenial,whose emphasis matches their interests, and whose price fits their budgets. The short introduc-torybackgroundsection in these Exercises , which precede each assignment, are intended only to fixnotation and provide official definitions and statements of important theorems for the exercisesand Problems which are a number of excellent online texts which are available free of charge. Among the bestareLinear Algebra [7] by Jim Hefferon, First Course in Linear Algebra [2] by Robert A. Beezer, very useful online resource is Przemyslaw Bogacki sLinear Algebra Toolkit[3]. ~bogacki/latAnd, of course, many topics in Linear Algebra are discussed with varying degrees of thoroughnessin theWikipedia[12] Eric Weisstein sMathworld[11].

6 The dozens and dozens of Linear Algebra books that have appeared, two that were writtenbefore dumbing down of textbooks became fashionable are especially notable, in my opinion,for the clarity of their authors mathematical vision: Paul Halmos sFinite-Dimensional VectorSpaces[6] and Hoffman and Kunze sLinear Algebra [8]. Some students, especially mathematicallyinclined ones, love these books, but others find them hard to read. If you are trying seriouslyto learn the subject, give them a look when you have the chance. Another excellent traditionaltext isLinear Algebra : An Introductory Approach[5] by Charles W. Curits. And for those moreinterested in applications bothElementary Linear Algebra : Applications Version[1] by HowardAnton and Chris Rorres andLinear Algebra and its Applications[10] by Gilbert Strang are loadedwith you are a student and find the level at which many of the current beginning Linear algebratexts are written depressingly pedestrian and the endless routine computations irritating, you mightexamine some of the more advanced texts.

7 Two excellent ones are Steven Roman sAdvanced LinearAlgebra[9] and William C. Brown sA Second Course in Linear Algebra [4].Concerning the material in these notes, I make no claims of originality. While I have dreamedup many of the items included here, there are many others which are standard Linear algebraexercises that can be traced back, in one form or another, through generations of Linear algebratexts, making any serious attempt at proper attribution quite futile. If anyone feels slighted, pleasecontact will surely be errors. I will be delighted to receive corrections, suggestions, or have placed the the LATEX source files on my web page so that those who wish to use these exer-cises for homework assignments, examinations, or any other noncommercial purpose can downloadthe material and, without having to retype everything, edit it and supplement it as they 1 MATRICES AND Linear EQUATIONSCHAPTER 1 SYSTEMS OF Linear BackgroundTopics:systems of Linear equations.

8 Gaussian elimination (Gauss method), elementary row op-erations, leading variables, free variables, echelon form, matrix, augmented matrix, Gauss-Jordanreduction, reduced echelon will say that an operation (sometimes calledscaling) which multiplies a rowof a matrix (or an equation) by a nonzero constant is arow operation of type I. An operation(sometimes calledswapping) that interchanges two rows of a matrix (or two equations) is arowoperation of type II. And an operation (sometimes calledpivoting) that adds a multiple of onerow of a matrix to another row (or adds a multiple of one equation to another) is arow operationof type SYSTEMS OF Linear Exercises (1) Suppose thatL1andL2are lines in the plane, that thex-intercepts ofL1andL2are 5and 1, respectively, and that the respectivey-intercepts are 5 and 1.

9 ThenL1andL2intersect at the point (,) .(2) Consider the following system of equations. w+x+y+z= 6w+y+z= 4w+y= 2( )(a) List the leading variables.(b) List the free variables.(c) The general solution of ( ) (expressed in terms of the free variables) is(,,,) .(d) Suppose that a fourth equation 2w+y= 5 is included in the system ( ). What isthe solution of the resulting system? Answer: (,,,).(e) Suppose that instead of the equation in part (d), the equation 2w 2y= 3 isincluded in the system ( ). Then what can you say about the solution(s) of theresulting system? Answer:.(3) Consider the following system of equations: x+y+z= 2x+ 3y+ 3z= 0x+ 3y+ 6z= 3( )(a) Use Gaussian elimination to put the augmented coefficient matrix into row echelonform.

10 The result will be 1 1 1a0 1 1b0 0 1c wherea=,b=, andc=.(b) Use Gauss-Jordan reduction to put the augmented coefficient matrix in reduced rowechelon form. The result will be 1 0 0d0 1 0e0 0 1f whered=,e=, andf=.(c) The solutions of ( ) arex=,y=, andz=.(4) Consider the following system of + (a) Using only row operation III and back substitution find the exact solution of thesystem. Answer:x=,y=.(b) Same as (a), but after performing each arithmetic operation round off your answer tofour significant figures. Answer:x=,y=. EXERCISES5(5) Find the values ofkfor which the system of equations{x+ky= 1kx+y= 1has (a) no solution. Answer:.(b) exactly one solution. Answer:.(c) infinitely many solutions .}