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
