the material,but it's only that I have trouble with the problems"reveals a lack of understanding of what we are up to.Being able to do things with the ideas is their point.The quotes below express this sentiment admirably.They state what I believe is the key to both the beauty and the power of mathematics and the sciences in general,and of linear algebra in particular(I took the liberty of formatting them as poems). I know of no better tactic than the illustration of exciting principles by well-chosen particulars. -Stephen Jay Gould If you really wish to learn then you must mount the machine and become acquainted with its tricks by actual trial. -Wilbur Wright Jim Hefferon Mathematics,Saint Michael's College Colchester,Vermont USA 05439 http://joshua.smcvt.edu 2006-May-20 Author's Note.Inventing a good exercise,one that enlightens as well as tests, is a creative act,and hard work.The inventor deserves recognition.But for some reason texts have traditionally not given attributions for questions.I have changed that here where I was sure of the source.I would greatly appreci- ate hearing from anyone who can help me to correctly attribute others of the questions. vi
the material, but it’s only that I have trouble with the problems” reveals a lack of understanding of what we are up to. Being able to do things with the ideas is their point. The quotes below express this sentiment admirably. They state what I believe is the key to both the beauty and the power of mathematics and the sciences in general, and of linear algebra in particular (I took the liberty of formatting them as poems). I know of no better tactic than the illustration of exciting principles by well-chosen particulars. –Stephen Jay Gould If you really wish to learn then you must mount the machine and become acquainted with its tricks by actual trial. –Wilbur Wright Jim Hefferon Mathematics, Saint Michael’s College Colchester, Vermont USA 05439 http://joshua.smcvt.edu 2006-May-20 Author’s Note. Inventing a good exercise, one that enlightens as well as tests, is a creative act, and hard work. The inventor deserves recognition. But for some reason texts have traditionally not given attributions for questions. I have changed that here where I was sure of the source. I would greatly appreciate hearing from anyone who can help me to correctly attribute others of the questions. vi
Contents Chapter One: Linear Systems 1 I Solving Linear Systems.················· 1 1 Gauss'Method..·················· 2 2 Describing the Solution Set....···..······ 3 General=Particular+Homogeneous。.··.······ 20 II Linear Geometry of n-Space···················· 32 1 Vectors in Space························· 32 2 Length and Angle Measures* 38 III Reduced Echelon Form.····:·················· 46 1 Gauss-Jordan Reduction.. 46 2 Row Equivalence.··.······················ Topic:Computer Algebra Systems 62 Topic:nput-Output Analysis.··· Topic:Accuracy of Computations 44 68 Topic:Analyzing Networks..... 72 Chapter Two:Vector Spaces 79 I Definition of Vector Space::···················· 80 1 Definition and Examples..,.·················· 80 2 Subspaces and Spanning Sets................... 91 IⅡLinear Independence·.. ...............101 1 Definition and Examples.·· 。 101 IⅡBasis and Dimension··..·· 112 1 Basis..··· 112 2 Dimension.。···。········· 。。 118 3 Vector Spaces and Linear Systems 123 4 Combining Subspaces*。··,··················· 130 Topic:Fields...·。..。.·。··。··。···。·。··· 140 Topic:Crystals::·::。···:·::。::·l42 Topic:Dimensional Analysis.················· ...146 vii
Contents Chapter One: Linear Systems 1 I Solving Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Gauss’ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Describing the Solution Set . . . . . . . . . . . . . . . . . . . . 11 3 General = Particular + Homogeneous . . . . . . . . . . . . . . 20 II Linear Geometry of n-Space . . . . . . . . . . . . . . . . . . . . . 32 1 Vectors in Space . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2 Length and Angle Measures∗ . . . . . . . . . . . . . . . . . . . 38 III Reduced Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . 46 1 Gauss-Jordan Reduction . . . . . . . . . . . . . . . . . . . . . . 46 2 Row Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Topic: Computer Algebra Systems . . . . . . . . . . . . . . . . . . . 62 Topic: Input-Output Analysis . . . . . . . . . . . . . . . . . . . . . . 64 Topic: Accuracy of Computations . . . . . . . . . . . . . . . . . . . . 68 Topic: Analyzing Networks . . . . . . . . . . . . . . . . . . . . . . . . 72 Chapter Two: Vector Spaces 79 I Definition of Vector Space . . . . . . . . . . . . . . . . . . . . . . 80 1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . 80 2 Subspaces and Spanning Sets . . . . . . . . . . . . . . . . . . . 91 II Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . 101 1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . 101 III Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 112 1 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3 Vector Spaces and Linear Systems . . . . . . . . . . . . . . . . 123 4 Combining Subspaces∗ . . . . . . . . . . . . . . . . . . . . . . . 130 Topic: Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Topic: Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Topic: Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . 146 vii
Chapter Three:Maps Between Spaces 153 I Isomorphisms..,.,:·.,...。.·。··.:..·:·.,153 1 Definition and Examples.···:················· 153 2 Dimension Characterizes Isomorphism 162 II Homomorphisms 170 2 Rangespace and Nullspace.·.·················· 177 IⅡComputing Linear Maps···· 189 1 Representing Linear Maps with Matrices... 189 2 Any Matrix Represents a Linear Map*········· 199 IV Matrix Operations...·.。。·..··········· 206 1 Sums and Scalar Products.··············· 206 2 Matrix Multiplication 4 208 3 Mechanics of Matrix Multiplication············· 216 4 Inverses.·.··········· 225 V Change of Basis。·..······················ 232 1 Changing Representations of Vectors.·············· 232 2 Changing Map Representations.:....。··.。.··.··· 236 VI Projection...........···. 244 1 Orthogonal Projection Into a Line 244 2 Gram-Schmidt Orthogonalization* ..·..248 3 Projection Into a Subspace*.··················· 254 Topic:Line of Best Fit.. 263 Topic:Geometry of Linear Maps 268 Topic:Markov Chains 。。。”··” 4。 275 Topic:Orthonormal Matrices 281 Chapter Four:Determinants 287 I Definition·········,··· 288 1 Exploration* 288 2 Properties of Determinants 4 293 3 The Permutation Expansion. 44 297 4 Determinants Exist*.,.·· 306 II Geometry of Determinants.... 。。·。。。。。。。。 313 1 Determinants as Size Functions············· 313 IⅡOther Formulas.........········ 320 1 Laplace's Expansion.·,,.。·.················ 320 Topic:Cramer's Rule..·····. 325 Topic:Speed of Calculating Determinants............... 328 Topic:Projective Geometry 331 Chapter Five:Similarity 343 I Complex Vector Spaces.···.:·.················ 343 1 Factoring and Complex Numbers:A Review* 344 2 Complex Representations 345 I Similarity....··.·....·..347 viii
Chapter Three: Maps Between Spaces 153 I Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . 153 2 Dimension Characterizes Isomorphism . . . . . . . . . . . . . . 162 II Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 2 Rangespace and Nullspace . . . . . . . . . . . . . . . . . . . . . 177 III Computing Linear Maps . . . . . . . . . . . . . . . . . . . . . . . 189 1 Representing Linear Maps with Matrices . . . . . . . . . . . . . 189 2 Any Matrix Represents a Linear Map∗ . . . . . . . . . . . . . . 199 IV Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 206 1 Sums and Scalar Products . . . . . . . . . . . . . . . . . . . . . 206 2 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . 208 3 Mechanics of Matrix Multiplication . . . . . . . . . . . . . . . . 216 4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 V Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 1 Changing Representations of Vectors . . . . . . . . . . . . . . . 232 2 Changing Map Representations . . . . . . . . . . . . . . . . . . 236 VI Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 1 Orthogonal Projection Into a Line∗ . . . . . . . . . . . . . . . . 244 2 Gram-Schmidt Orthogonalization∗ . . . . . . . . . . . . . . . . 248 3 Projection Into a Subspace∗ . . . . . . . . . . . . . . . . . . . . 254 Topic: Line of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Topic: Geometry of Linear Maps . . . . . . . . . . . . . . . . . . . . 268 Topic: Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Topic: Orthonormal Matrices . . . . . . . . . . . . . . . . . . . . . . 281 Chapter Four: Determinants 287 I Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 1 Exploration∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 2 Properties of Determinants . . . . . . . . . . . . . . . . . . . . 293 3 The Permutation Expansion . . . . . . . . . . . . . . . . . . . . 297 4 Determinants Exist∗ . . . . . . . . . . . . . . . . . . . . . . . . 306 II Geometry of Determinants . . . . . . . . . . . . . . . . . . . . . . 313 1 Determinants as Size Functions . . . . . . . . . . . . . . . . . . 313 III Other Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 1 Laplace’s Expansion∗ . . . . . . . . . . . . . . . . . . . . . . . . 320 Topic: Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Topic: Speed of Calculating Determinants . . . . . . . . . . . . . . . 328 Topic: Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . 331 Chapter Five: Similarity 343 I Complex Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 343 1 Factoring and Complex Numbers; A Review∗ . . . . . . . . . . 344 2 Complex Representations . . . . . . . . . . . . . . . . . . . . . 345 II Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 viii
1 Definition and Examples ... 347 2 Diagonalizability.,·。·.·.·················· 349 3 Eigenvalues and Eigenvectors··· 353 III Nilpotence 361 1 Self-Composition" 361 2 Strings*。364 V Jordan Form..,..··。,。·。············ 375 1 Polynomials of Maps and Matrices◆.··············· 375 2 Jordan Canonical Form*......·....。...· 。。。。。 382 Topic:Method of Powers.························ 395 Topic:Stable Populations.·..·.·.·.·.··.···.····· 399 Topic:Linear Recurrences 401 Appendix A-1 Propositions ........ ·。·.........。...A-1 Quantifiers 4 ····.A-3 Techniques of Proof·.······· 4。 A-5 Sets.Functions,and Relations ... A-7 *Note:starred subsections are optional. 这
1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . 347 2 Diagonalizability . . . . . . . . . . . . . . . . . . . . . . . . . . 349 3 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . 353 III Nilpotence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 1 Self-Composition∗ . . . . . . . . . . . . . . . . . . . . . . . . . 361 2 Strings∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 IV Jordan Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 1 Polynomials of Maps and Matrices∗ . . . . . . . . . . . . . . . . 375 2 Jordan Canonical Form∗ . . . . . . . . . . . . . . . . . . . . . . 382 Topic: Method of Powers . . . . . . . . . . . . . . . . . . . . . . . . . 395 Topic: Stable Populations . . . . . . . . . . . . . . . . . . . . . . . . 399 Topic: Linear Recurrences . . . . . . . . . . . . . . . . . . . . . . . . 401 Appendix A-1 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3 Techniques of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . A-5 Sets, Functions, and Relations . . . . . . . . . . . . . . . . . . . . . A-7 ∗Note: starred subsections are optional. ix
Chapter One Linear Systems I Solving Linear Systems Systems of linear equations are common in science and mathematics.These two examples from high school science [Onan]give a sense of how they arise. The first example is from Physics.Suppose that we are given three objects, one with a mass known to be 2 kg.and are asked to find the unknown masses. Suppose further that experimentation with a meter stick produces these two balances 一40 —50- 25十 一50 @(c) 2 (②① 15 25州 Since the sum of moments on the left of each balance equals the sum of moments on the right (the moment of an object is its mass times its distance from the balance point),the two balances give this system of two equations. 40h+15c=100 25c=50+50h The second example of a linear system is from Chemistry.We can mix, under controlled conditions,toluene C7Hs and nitric acid HNO3 to produce trinitrotoluene C7H5O6N3 along with the byproduct water(conditions have to be controlled very well,indeed-trinitrotoluene is better known as TNT).In what proportion should those components be mixed?The number of atoms of each element present before the reaction x C7Hs yHNO3 -zC7H506N3 wH2O must equal the number present afterward.Applying that principle to the ele- 1
Chapter One Linear Systems I Solving Linear Systems Systems of linear equations are common in science and mathematics. These two examples from high school science [Onan] give a sense of how they arise. The first example is from Physics. Suppose that we are given three objects, one with a mass known to be 2 kg, and are asked to find the unknown masses. Suppose further that experimentation with a meter stick produces these two balances. c h 2 15 40 50 c 2 h 25 50 25 Since the sum of moments on the left of each balance equals the sum of moments on the right (the moment of an object is its mass times its distance from the balance point), the two balances give this system of two equations. 40h + 15c = 100 25c = 50 + 50h The second example of a linear system is from Chemistry. We can mix, under controlled conditions, toluene C7H8 and nitric acid HNO3 to produce trinitrotoluene C7H5O6N3 along with the byproduct water (conditions have to be controlled very well, indeed— trinitrotoluene is better known as TNT). In what proportion should those components be mixed? The number of atoms of each element present before the reaction x C7H8 + y HNO3 −→ z C7H5O6N3 + w H2O must equal the number present afterward. Applying that principle to the ele- 1