本书旨在弥合主要面向计算的低年级本科课程与高级数学课程中遇到的抽象数学之间的差距。
3.1 The Fundamental Theorem of Algebra And Factoring Polynomials 3 The fundamental theorem of algebra 26 3.2 Factoring polynomials Exercises 4 Vector Spaces 36 4.1 Definition of vector spaces 36 4.2 Elementary properties of vector spaces 4.3 Subspaces 4.4 Sums and direct sums Exercises 5 Span and bases 5.1 Linear span 48 5.2 Linear independence 5.3 bases 5. 4 Dimension Exercises 6 Linear Maps ⑥64 6.1 Definition and elementary properties 6.2 Null spaces 6.3 Range 6.4 Homomorphisms 6.5 The dimension formula 6.6 The matrix of a linear map 6.7 Invertibility E Cerises 7 Eigenvalues and Eigenvectors 7.1 Invariant subspaces 7. 2 Eigenvalues 86 7. 3 Diagonal matrices 7.4 Existence of eigenvalues 7.5 Upper triangular matrices 7.6 Diagonalization of 2 x 2 matrices and applications Exercises 8 Permutations and the Determinant of a Square matrix 102 8.1 Permutations 8. 1.1 Definition of permutations 山O2 8.1.2 Composition of permutations LO⑤ 8.1.3 Inversions and the sign of a permutation I07 8.2 Determinants 110 8.2.1 Summations indexed by the set of all permutations 8.2.2 Properties of the determinant 8.2.3 Further properties and applications 8.2.4 Computing determinants with cofactor expansions Exercises 8 9 Inner Product Spaces I20 9.1 Inner product. 9.2 Norms I22 9.3 Orthogonality T24 9.4 Orthonormal bases 9.5 The Gran-SchInidt orthogonalization procedure 9.6 Orthogonal projections and minimization problems 132 Exercises I36 10 Change of Bases 139 10.1 Coordinate vectors 10.2 Change of basis transformation 141 Exercises 145 11 The Spectral Theorem for Normal Linear Maps 147 11.1 Self-adjoint or hermitian operators 11.2 Normal operators 49 11.3 Normal operators and the spectral decomposition IV 11.4 Applications of the Spectral Theorem: diagonalization 11.5 Positive operators 57 11.6 Polar decomposition 11.7 Singular-value decomposition Exercises 61 List of appendices A Supplementary Notes on Matrices and Linear Systems A 1 From linear systems to matrix equations A 1. 1 Definition of and notation for matrices I65 A.1.2 Using matrices to encode linear systems A 2 Matrix arithmetic A 2.1 Addition and scalar multiplication A.2.2 Multiplication of matrices I7⑤ A.2. 3 Invertibility of square matrices A3 Solving linear systems by factoring the coefficient matrix A.3. 1 Factorizing matrices using Gaussian elimination A.3.2 Solving homogeneous linear systems A 3.3 Solving inhomogeneous linear systems 195 A.3.4 Solving linear systems with LU-factorization 99 A 4 Matrices and linear maps ......2O4 4.1 The canonical matrix of a linear map A 4.2 Using linear maps to solve linear systems 205 A.5 Special operations on matrices 2 A.5.1 Transpose and conjugate transpose A.5.2 The trace of a square matrix Exercises 2I4 b The language of sets and functions B. 1 Sets 2I8 B2 Subset, uniOn, intersection, and Cartesian product 220 B 3 Relations B 4 Functions 223 c Summary of algebraic Structures Encountered 226 C. 1 Binary operations and scaling operations 26 C2 Groups. fields, and vector spaces C 3 Rings and algebras 233 D Some Common math Symbols and abbreviations 23 E Summary of Notation Used 248 F Movie Scripts 246 Chapter 1 What is Linear Algebra? 1.1 Introduction This book aims to bridge the gap between the mainly computation-oriented lower division undergraduate classes and the abstract mathematics encountered in more advanced mathe- matics courses. The goal of this book is threefold 1. You will learn Lincar Algebra, which is one of the most widely used mathematical theories around. Linear Algebra finds applications in virtually every area of mathe matics, including multivariate calculus, differential equations, and probability theory It is also widely applied in fields like physics, chemistry, economics, psychology, and engineering. You are even relying on methods from Linear Algebra every time you use an internet search like Google, the Global Positioning System(GPS), or a cellphone 2. You will acquire computational skills to solve linear systems of equations, perform operations on matrices, calculate eigenvalues and find determinants of matrices 3. In the setting of Linear algebra, you will be introduced to abstraction. As the theory of Linear algebra is developed, you will learn how to make and use definitions and how to write proofs The exercises for each Chapter are divided into nore computation-oriented exercises and exercises that focus on proof-writing 2 CHAPTER 1. WHAT IS LINEAR ALGEBRA? 1.2 What is Linear algebra? Linear Algebra is the branch of mathematics aimed at solving systems of linear equations with a finite number of unknowns. In particular, one would like to obtain answers to the llowing questions Characterization of solutions: Are there solutions to a given system of linear equations? How many solutions are there Finding solutions: How does the solution set look? What are the solutions? Linear algebra is a systematic theory regarding the solutions of systems of linear equations Example 1.2.1. Let us take the following system of two linear equations in the two un- knowns 1 and 2x1+x2=0 This system has a unique solution for a1, 2 CR, namely ai=, and C2=2 The solution can be found in several different ways. One approach is to first solve for one of the unknowns in one of the equations and then to substitute the result into the other quation. Here. for example, we might solve to obtain 1+x2 from the second equation. Then, substituting this in place of l in the first equation, we Lve 2(1+x2)+2=0 From this, 2=-2/3. Then, by further substitution x1=1+ Alternatively, we can take a more systematic approach in eliminating variables. Here for example, we can subtract 2 times the second equation from the first equation in order to obtain 3. 2 =-2. It is then innediate that u 2=-= and, by substituting this value for 2 in the first equation, that 01-3 1.2. WHAT IS LINEAR ALGEBRA? 3 Example 1.2.2. Take the following system of two linear equations in the two unknowns x1 and 1+2 2x1+2 We can eliminate variables by adding -2 times the first equation to the second equation which results in o This is obviously a contradiction, and hence this system of equations has no solution Example 1. 2.3. Let us take the following system of one linear equation in the two unknowns CI and 2: C In this case, there are infinitely many solutions given by the set. [2=301 T1ER. You can think of this solution set as a line in the Euclidean plane r 1 In general, a system of m linear equations in n unknowns T1, T2,..., In is a collec tion of equations of the form 113C1+a12x℃2+… 211+222+…+a b2 am11+am22+ + mnen where the aii 's are the coefficients(usually real or complex numbers)in front of the unknowns and the bi's are also fixed real or complex numbers. A solution is a set of numbers $1,$2,..., Sn such that, substituting C1=$1, I2=$2, .. Cn=Sn for the unknowns. all of the equations in System (1. 1b hold. Linear Algebra is a theory that concerns the solutions and the structure of solutions for linear equations. As we progress, you will see that there a lot of subtlety in fully understanding the solutions for such equations 4 CHAPTER 1. WHAT IS LINEAR ALGEBRA? 1.3 Systems of linear equations 1.3.1 Linear equations Before going on, let us reformulate the notion of a system of linear equations into the Language of functions. This will also help us understand the adjective "linear"a bit better nction a nap f:X→Y from a set x to a set y The set x is called the domain of the function, and the set y is called the target space or codomain of the function. An equation is where E X and y E Y.(If you are not familiar with the abstract notions of sets and functions, please consult Appendix Bl) Example 1.3.1. Let f R-R be the function f(r)=r3-T. Then f(ar)=3-T=1 is an equation. The domain and target space are both the set of real numbers R in this case In this setting, a system of equations is just another kind of equation Example 1.3.2. Let X=Y= R2=RXR be the Cartesian product of the set of real numbers. Then define the function f: R2 R2 as f(x1,x2)=(2x1+x2,x1-x2) and set y=(0, 1). Then the equation f(a)=y, where =(21, 2)C R, describes the system of linear equations of Example 1. 2.11 The next question we need to answer is, "What is a linear equation? Building on the definition of an equation, a linear equation is any equation defined by a "linear"'function f that is defined on a"linear"space(a k.a. a vector space as defined in Section will elaborate on all of this in later chapters, but let us deMonstrate the nain features of a "linear"space in terms of the example R2. Take -(31, 2),y=(1, 32)E R2. There are