Linear Algebra A First Course with Applications to Differential Equations

by Apostol, Tom M.

• ISBN13: 9780471174219
• ISBN10: 0471174211
• Edition: 1st
• Format: Hardcover
• Copyright: 1997-07-15
• Publisher: Wiley-Interscience
• More Book Details
• Purchase Benefits

Summary

Developed from the author's successful two-volume Calculus text this book presents Linear Algebra without emphasis on abstraction or formalization. To accommodate a variety of backgrounds, the text begins with a review of prerequisites divided into precalculus and calculus prerequisites. It continues to cover vector algebra, analytic geometry, linear spaces, determinants, linear differential equations and more.

Author Biography

TOM M. APOSTOL, Emeritus Professor at the California Institute of Technology, is the author of several highly regarded texts on calculus, analysis, and number theory, and is Director of Project MATHEMATICS!, a series of computer-animated mathematics videotapes.

Table of Contents

Preface	xv
0. REVIEW OF PREREQUISITES	1	(24)
Part 1. Pre-calculus Prerequisites	1	(13)
0.1 Real numbers as points on a line	1	(1)
0.2 Pairs of real numbers as points in a plane	2	(1)
0.3 Polar coordinates	3	(1)
0.4 Complex numbers	4	(1)
0.5 Definition and algebraic properties of complex numbers	5	(1)
0.6 Complex numbers as an extension of real numbers	6	(1)
0.7 The imaginary unit i	6	(2)
0.8 Exercises	8	(1)
0.9 Geometric interpretation. Modulus and argument	8	(2)
0.10 Complex conjugates	10	(1)
0.11 Exercises	10	(1)
0.12 Mathematical induction	11	(2)
0.13 Exercises	13	(1)
0.14 Necessary and sufficient	13	(1)
Part 2. Calculus Prerequisites	14	(11)
0.15 The concept of derivative	14	(1)
0.16 Basic properties of derivatives	15	(1)
0.17 Derivatives of some elementary functions	16	(1)
0.18 Velocity and acceleration	16	(1)
0.19 The area problem and the history of integral calculus	17	(1)
0.20 Integration as a process for producing new functions	18	(1)
0.21 Basic properties of the integral	18	(1)
0.22 The exponential function	19	(1)
0.23 Complex exponentials	20	(1)
0.24 Polar form of complex numbers	21	(1)
0.25 Power series and series of functions	22	(2)
0.26 Exercises	24	(1)
1. VECTOR ALGEBRA	25	(26)
1.1 Historical introduction	25	(1)
1.2 The vector space of n-tuples of real numbers	26	(2)
1.3 Geometric interpretation for n less than 3	28	(2)
1.4 Exercises	30	(1)
1.5 The dot product	31	(2)
1.6 Length or norm of a vector	33	(2)
1.7 Orthogonality of vectors	35	(1)
1.8 Exercises	36	(1)
1.9 Projections. Angle between vectors in n-space	37	(2)
1.10 The unit coordinate vectors	39	(2)
1.11 Exercises	41	(1)
1.12 The linear span of a finite set of vectors	42	(1)
1.13 Linear independence	43	(3)
1.14 Bases	46	(1)
1.15 Exercises	47	(1)
1.16 The vector space Cn of n-tuples of complex numbers	48	(2)
1.17 Exercises	50	(1)
2. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY	51	(40)
2.1 Introduction	51	(11)
2.2 Lines in n-space	52	(1)
12.3 Some simple properties of straight lines in Rn	53	(1)
2.4 Lines and vector-valued functions in n-space	54	(1)
2.5 Lines in 3-space in 2-space	55	(2)
2.6 Exercises	57	(1)
2.7 Planes in Euclidean n-space	58	(3)
2.8 Planes and vector-valued functions	61	(1)
2.9 Exercises	62	(1)
2.10 The cross product of two vectors in R3	63	(3)
2.11 The cross product expressed as a determinant	66	(1)
2.12 Exercises	67	(1)
2.13 The scalar triple product	68	(2)
2.14 Cramer's rule for solving a system of three linear equations	70	(1)
2.15 Exercises	71	(2)
2.16 Normal vectors to planes in R3	73	(1)
2.17 Linear Cartesian equations for planes in R3	74	(2)
2.18 Exercises	76	(1)
2.19 The conic sections	77	(2)
2.20 Eccentricity of conic sections	79	(2)
2.21 Polar equations for conic sections	81	(1)
2.22 Exercises	82	(1)
2.23 Cartesian equation for a general conic	83	(1)
2.24 Conic sections symmetric about the origin	84	(1)
2.25 Cartesian equations for the ellipse and the hyperbola in standard position	85	(2)
2.26 Cartesian equations for the parabola	87	(1)
2.27 Exercises	88	(2)
2.28 Miscellaneous exercises on conic sections	90	(1)
3. LINEAR SPACES	91	(28)
3.1 Introduction	91	(1)
3.2 Axiomatic definition of a linear space	91	(1)
3.3 Examples of linear spaces	92	(2)
3.4 Elementary consequences of the axioms	94	(1)
3.5 Exercises	95	(1)
3.6 Subspaces of a linear space	96	(2)
3.7 Dependent and independent sets in a linear space	98	(2)
3.8 Bases and dimension	100	(2)
3.9 Components	102	(1)
3.10 Exercises	102	(1)
3.11 Inner products, Euclidean spaces. Norms	103	(4)
3.12 Orthogonality in a Euclidean space	107	(2)
3.13 Exercises	109	(2)
3.14 Construction of orthogonal sets. The Gram-Schmidt process	111	(4)
3.15 Orthogonal complements. Projections	115	(1)
3.16 Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace	116	(2)
3.17 Exercises	118	(1)
4. LINEAR TRANSFORMATIONS AND MATRICES	119	(42)
4.1 Linear transformations	119	(2)
4.2 Null space and range	121	(1)
4.3 Nullity and rank	122	(1)
4.4 Exercises	123	(1)
4.5 Algebraic operations on linear transformations	124	(2)
4.6 Inverses	126	(2)
4.7 One-to-one linear transformations	128	(2)
4.8 Exercises	130	(1)
4.9 Linear transformations with prescribed values at the elements of a basis	131	(1)
4.10 Matrix representations of linear transformations	132	(5)
4.11 Construction of a matrix representation in diagonal form	137	(2)
4.12 Exercises	139	(1)
4.13 Linear spaces of matrices	140	(2)
4.14 Isomorphism between linear transformations and matrices	142	(1)
4.15 Multiplication of matrices	143	(3)
4.16 Exercises	146	(1)
4.17 Applications to systems of linear equations	147	(3)
4.18 Computation techniques. The Gauss-Jordan method	150	(4)
4.19 Inverses of square matrices	154	(4)
4.20 Exercises	158	(1)
4.21 Miscellaneous exercises on matrices	159	(2)
5. DETERMINANTS	161	(26)
5.1 Introduction	161	(1)
5.2 Motivation for the choice of axioms for a determinant function	162	(1)
5.3 A set of axioms for a determinant function	163	(1)
5.4 The determinant of a diagonal matrix	164	(1)
5.5 The determinant of an upper triangular matrix	165	(1)
5.6 Computation of any determinant by the Gauss-Jordan process	166	(1)
5.7 Uniqueness of the determinant function	166	(1)
5.8 Exercises	167	(1)
5.9 Multilinearity of determinants	168	(2)
5.10 Applications of multilinearity	170	(1)
5.11 The product formula for determinants	171	(1)
5.12 The determinant of the inverse of a nonsingular matrix	172	(1)
5.13 Determinants and independence of vectors	173	(1)
5.14 The determinant of a block-diagonal matrix	173	(1)
5.15 Exercises	174	(1)
5.16 Expansion formulas by cofactors	175	(2)
5.17 The cofactor matrix	177	(1)
5.18 Cramer's rule	178	(1)
5.19 Expansion formulas by minors	179	(2)
5.20 Exercises	181	(1)
5.21 Existence of the determinant function	182	(2)
5.22 Miscellaneous exercises on determinants	184	(3)
6. EIGENVALUES AND EIGENVECTORS	187	(30)
6.1 Linear transformations with diagonal matrix representations	187	(1)
6.2 Eigenvalues and eigenvectors of a linear transformation	188	(2)
6.3 Linear independence of eigenvectors corresponding to distinct eigenvalues	190	(1)
6.4 Exercises	191	(1)
6.5 The finite-dimensional case	192	(1)
6.6 The triangularization theorem	193	(3)
6.7 Characteristic polynomials	196	(2)
6.8 Calculation of eigenvalues and eigenvectors in the finite-dimensional case	198	(3)
6.9 The product and sum of the roots of a characteristic polynomial	201	(1)
6.10 Exercises	202	(1)
6.11 Matrices representing the same linear transformation. Similar matrices	203	(4)
6.12 Exercises	207	(1)
6.13 The Cayley-Hamilton theorem	208	(3)
6.14 Exercises	211	(1)
6.15 The Jordan normal form	211	(3)
6.16 Miscellaneous exercises on eigenvalues and eigenvectors	214	(3)
7. EIGENVALUES OF OPERATORS ACTING ON EUCLIDEAN SPACES	217	(30)
7.1 Eigenvalues and inner products	217	(1)
7.2 Hermitian and skew-Hermitian transformations	218	(1)
7.3 Orthogonality of eigenvectors corresponding to distinct eigenvalues	219	(1)
Exercises	219	(1)
7.5 Existence of an orthonormal set of eigenvectors for Hermitian and skew-Hermitian operators acting on finite-dimensional spaces	220	(1)
7.6 Matrix representations for Hermitian and skew-Hermitian operators	221	(1)
7.7 Hermitian and skew-Hermitian matrices. The adjoint of a matrix	222	(1)
7.8 Diagonalization of a Hermitian or skew-Hermitian matrix	223	(1)
7.9 Unitary matrices. Orthogonal matrices	224	(1)
7.10 Exercises	225	(2)
7.11 Quadratic forms	227	(2)
7.12 Reduction of a real quadratic form to a diagonal form	229	(2)
7.13 Applications to conic sections	231	(4)
7.14 Exercises	235	(1)
7.15 Positive definite quadratic forms	235	(1)
*7.16 Eigenvalues of a symmetric transformation obtained as values of its quadratic form	236	(2)
*7.17 Extremal properties of eigenvalues of a symmetric transformation	238	(1)
*7.18 The finite-dimensional case	239	(1)
7.19 Unitary transformations	240	(2)
7.20 Exercises	242	(1)
*7.21 Example of symmetric and skew-symmetric operators acting on function spaces	243	(2)
7.22 Exercises	245	(2)
8. APPLICATIONS TO LINEAR DIFFERENTIAL EQUATIONS	247	(26)
8.1 Introduction	247	(1)
8.2 Review of results concerning linear differential equations of first and second orders	248	(2)
8.3 Exercises	250	(1)
8.4 Linear differential of order n	251	(1)
8.5 The existence-uniqueness theorem	252	(1)
8.6 The dimension of the solution space of a homogeneous linear differential equation	252	(1)
8.7 The algebra of constant-coefficient operators	253	(2)
8.8 Determination of a basis of solutions for linear equations with constant coefficients by factorization of operators	255	(4)
Exercises	259	(1)
8.10 The relation between the homogeneous and nonhomogeneous equations	260	(1)
8.11 Determination of a particular solution of the nonhomogeneous equation. The method of variation of parameters	261	(4)
8.12 Nonsingularity of the Wronskian matrix of n independent solutions of a homogeneous linear equation	265	(1)
8.13 Special methods for determining a particular solution of the nonhomogeneous equation. Reduction to a system of first-order linear equations	266	(1)
8.14 The annihilator method for determining a particular solution of the nonhomogeneous equation	267	(3)
8.15 Exercises	270	(3)
9. APPLICATIONS TO SYSTEMS OF DIFFERENTIAL EQUATIONS	273	(30)
9.1 Introduction	273	(2)
9.2 Calculus of matrix functions	275	(1)
9.3 Infinite series of matrices. Norms of matrices	276	(1)
9.4 Exercises	277	(1)
9.5 The exponential matrix	278	(1)
9.6 The differential equation satisfied by etA	279	(1)
9.7 Uniqueness theorem for the matrix differential equation F1(t) = AF(t)	279	(2)
9.8 The law of exponents for exponential matrices	281	(1)
9.9 Existence and uniqueness theorems for homogeneous linear systems with constant coefficient	281	(1)
9.10 Calculating etA in special cases	282	(5)
9.11 Exercises	287	(1)
9.12 Putzer's method for calculating etA	288	(3)
9.13 Alternate methods for calculating etA in special cases	291	(2)
9.14 Exercises	293	(1)
9.15 Nonhomogeneous linear systems with constant coefficients	294	(2)
9.16 Exercises	296	(2)
9.17 The general linear system Y1(t) = P(t)Y(t) + Q(t)	298	(3)
9.18 A power series method for solving homogeneous linear systems	301	(1)
9.19 Exercises	302	(1)
10. THE METHOD OF SUCCESSIVE APPROXIMATIONS	303	(18)
10.1 Introduction	303	(1)
10.2 Application to the homogeneous linear system Y1(t) = A(t)Y(t)	303	(1)
10.3 Convergence of the sequence of successive approximations	304	(4)
10.4 The method of successive approximations applied to first-order nonlinear systems	308	(1)
10.5 Proof of an existence-uniqueness theorem for first-order nonlinear systems	309	(2)
10.6 Exercises	311	(1)
*10.7 Successive approximations and fixed points of operators	312	(1)
*10.8 Normed linear spaces	313	(1)
*10.9 Contraction operators	314	(1)
*10.10 Fixed-point theorem for contraction operators	315	(2)
*10.11 Applications of the fixed-point theorem	317	(4)
Answers to Exercises	321	(22)
Index	343