핵심선형 대수학(개정판 3판)

서론
Acronyms and Key notations

1 Systems of linear equations with Matrices
1.1 From Nine Chapters on the Mathematical Art
1.2 Three types of solution sets of systems of linear equations
1.3 Elementary operations; Eliminations and substitutions
1.4 Gauss-Jordan elimination
1.5 Matrices; Their sums and scalar multiplications
1.6 Products of vectors and products of matrices
1.7 Inverse matrices; One-sided and two-sided inverses
1.8 Elementary matrices for Gauss-Jordan elimination
1.9 When and how can one find the inverse ofatrix?
1.10 Applications
1.10.1 Permutation matrices
1.10.2 Sums and products of block matrices
1.10.3 How to use LU decompositions
1.11 Exercises

2 Determinants; The signed areas and the signed volumes
2.1 Leibniz's letter to de l'H^opital
2.2 Areas of parallelograms and determinants
2.3 Definition and properties of the determinant
2.4 The determinant function detis well defined
2.5 Cofactor expansions make it easy to compute the determinant
2.6 Cramer's rule for solvingystem of linear equations
2.7 Applications
2.7.1 Determinants of Vandermonde and circulant matrices
2.7.2 Exact polynomial curve fitting to data
2.7.3 Cauchy-Binet formula to evaluate the determinant det(AB)
2.7.4 Areas and volumes by determinants
2.8 Exercises

3 Vector Spaces; Abstraction of the plane R2 and the space R3
3.1 From the 3-space R3 to vector spaces
3.2 Creating new vector spaces from old ones
3.2.1 Which subsets can be vector spaces?
3.2.2 Sums and intersections of subspaces
3.2.3 Subspace spanned byet
3.3 Linear dependence and Linear independence
3.4 Bases and coordinate systems for vector spaces
3.5 Every vector space has its dimension
3.6 Always one can constructasis forector space
3.7 To every matrix, there are Row, Column, and Null spaces
3.8 Rank-Nullity theorem; Relation between rank and nullity
3.9 Applications
3.9.1 One-sided or two-sided invertibility
3.9.2 Rank decompositions
3.10 Exercises

4 Linear Transformations; Linearity preserving functions
4.1 Examples and properties of linear transformations
4.2 Constructing linear transformations
4.3 Which linear transformation has the inverse linear transformation?
4.4 Natural isomorphisms and coordinate systems for vector space
4.5 Matrix representations of linear transformations
4.6 Change of bases; Change of coordinates
4.7 Which two matrices are similar?
4.8 Applications
4.8.1 Affine spaces and affine transformations
4.8.2 Vector spaces of linear transformations
4.8.3 Homogeneous coordinate systems and graphics
4.9 Exercises

5 Inner Product Spaces; Abstraction of Euclidean spaces
5.1 Inner products; Abstraction of the dot product
5.2 Geometry of vectors; Lengths, angles, orthogonality
5.3 Matrix representations of inner products
5.4 Gram-Schmidt orthogonalization for rectangular coordinates
5.5 Projections ofector space and their algebraic characterizatio
5.6 Orthogonal and oblique projections of an inner product space
5.7 Orthogonal projection matrices and Oblique projection matrice
5.8 Orthogonal complements of the row and the column spaces
5.9 Orthogonal matrices preserve the lengths, so are isometries
5.10 Applications
5.10.1 Least squares solutions ofystem Ax=balways exis
5.10.2 Least squares solutions of Ax=bwhen Ais of full rank
5.10.3 Best polynomial curve fitting using least squares metho
5.10.4 What is the entire class of orthogonal projection matrices?
5.10.5 QR decompositions from the Gram-Schmidt process
5.11 Exercises

6 Diagonalization makes it easy, as in Powers of Matrices
6.1 Eigenvalues and eigenvectors
6.2 Properties of eigenvalues and eigenvectors
6.3 Diagonalizations and Eigenvalue decompositions(EVD)
6.4 Applications
6.4.1 Fibonacci sequence
6.4.2 Linear recurrence relations; LRR
6.4.3 Linear difference equations; LdE
6.4.4 Discrete dynamical systems
6.4.5 Markov processes
6.4.6 Linear differential equations; LDE
6.4.7 LDE y0=Ayforiagonalizable matrix A.
6.5 Exponential matrices
6.6 Applications continued
6.6.1 Existence and uniqueness of solutions to an LDE
6.7 Exercises

7 Complex Vector Spaces; Abstraction of the complex plane
7.1 How are complex vector spaces different from real vector spaces?
7.2 Hermitian inner products
7.3 Orthogonal projections, the complex case
7.4 Hermitian, skew-Hermitian, and unitary matrices
7.5 Hermitian, skew-Hermitian, and unitary matrices are all unitarily diagonalizable
7.6 Which real matrices are all of orthogonally diagonalizable?
7.7 Which complex matrices are all of unitarily diagonalizable?
7.8 Applications
7.8.1 The eigenvalue decompositions(EVD) of normal matrices and computing the power Ak
7.8.2 Singular value decompositions(SVD); Diagonalizations of rectangular matrices
7.8.3 How to findne-sided (left or right) inverse ofatri
7.9 Exercises

8 Jordan Canonical Forms; Block diagonalizations of Matrices
8.1 Jordan blocks and Jordan matrices
8.2 Jordan canonical forms and Jordan decompositions
8.3 How to find the Jordan canonical form J.
8.4 How to findhange of basis matrix Qforordan decomposition
8.5 Computing Jk and eJ forordan matrix J
8.6 Cayley-Hamilton theorem
8.7 Computing the power Ak by using the Cayley-Hamilton theore
8.8 Applications
8.8.1 The minimal polynomial ofatrix
8.8.2 Computing Ak by using the minimal polynomial
8.8.3 Computing eA by using the minimal polynomial
8.8.4 LdE xn=Axn-1for any square matrix A.
8.8.5 LDE y0=Ayfor any square matrix A.
8.9 Exercises

9 Quadratic equations and Quadratic Forms with Matrices
9.1 Can you solve (multivariate) quadratic equations?
9.2 Quadratic forms; q(x)=xHAxwith symmetric or Hermitian A.
9.3 Classification of quadratic curves and quadratic surfaces
9.4 Diagonalization of quadratic forms
9.5 The inertia classifies level surfaces
9.6 Characterizations of definite forms
9.7 Change of bases for quadratic forms and the congruence relatio
9.8 Computing the inertia using the congruence relation
9.9 Bilinear and sesquilinear forms; b(x;y)=xHAy.
9.10 Diagonalization of symmetric bilinear or Hermitian sesquilinear forms
9.11 Applications
9.11.1 Nondegenerate bilinear forms
9.11.2 Sylvester's law of inertia
9.11.3 Real quadratic forms versus symmetric bilinear forms
9.11.4 MinMax of quadratic forms and estimate eigenvalues
9.12 Exercises

Selected Answers and Hints
Bibliography
Index

곽진호 중국 정부의 천인계획에 따른 북경교통대학 수학과와 한능신능원학원(漢能新能源學院) 소속(설립원장) 중국정부 특별초빙 교수이다. 연구분야는 계수(計數)적 조합론과 조합적 위상수학으로, 130여 편의 논문을 발표했으며 400여 명의 국제 수학자들이 그 의 논문을 인용하고 있다. 2005-2010년에 엘스비어(Elsevier) 출판사로부터 Top-Cited 논문상을 수상하기도 하였으며, 현재 여러 국제학술지 편집위원으로 활동하고 있다. 또한 다수의 국내·외 저서를 집필하였고, 포항공과대학교의 개교 준비교수 중 한 사람으로서 한국연구재단 우수연구센터 사업에 19년간 참여해 왔다(Com2MaC 소장). 그 결과 국제공동연구팀들이 구성한 국제학술회의는 현재 매년 일회씩 슬로베니아와 러시아 노보시비리스크에서 각각 실시되고 있으며 국제 공인 유명학회가 되었다. 미국 인디애나대학교에서 박사학위를 취득(1982년)하고, 미국 미시건대학교(1985-1986년), 영국 케임브리지대학교(1992년), 미국 컬럼비아대학교(1997년), 중국 칭화대학교(2004년), 독일 아헨공과대학교(2017년) 등에 방문 교수로 연구와 교육에 참여하였다. 한편 한국 정부에서 옥조근정훈장을, 중국 정부에서 외국인에게 수여하는 최고 훈장인 중국정부우의상(中國政府友誼奬)을 수상하였다. 재중한인과학기술자협회 설립회장(2015년)이며 한중기초과학교류위원회 공동위원장을 역임하였고 포항공과대학교 명예교수이다