핵심 선형 대수학
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9791160736243
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| Publication Date | 2023/03/02 |
| Pages/Weight/Size | 188*257*35mm |
| ISBN | 9791160736243 |
| Categories | 대학교재 > 자연과학 계열 |
Description
이 책은 1997년과 2004년에 Birkhauser에서 출간된 『선형대수학』(S. Hong 공저)에 뿌리를 두고 있다. 새로운 출판을 위하여 미흡했던 내용들을 보완하여 추가하고 기존 내용들을 보다 쉽게 재정리 하였다. 한 거의 모든 예제와 연습문제를 독자들의 이해를 넓히고 흥미를 유도할 수 있도록 새롭게 정리하고, 집합 혹은 함수와 같은 수학의 기본 개념과 기초적인 수학 표기법은 독자들의 기억을 되살리기 위해 각주에 간단히 기술하였다. 선형대수학 발전에 기여한 수학자들의 역사적인 업적과 약력도 독자들이 과학과 수학의 역사적 발전 추세를 읽을 수 있도록 각주에 추가하였다.
Contents
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
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