서론 iii Acronyms and Key notations v 1 Systems of linear equations and LU decompositions 1 1.1 From the Nine Chapters on the Mathematical Art . 1 1.2 Three types of solution sets of systems of linear equations . 2 1.3 Elementary operations; Eliminations and substitutions .  6 1.4 Gauss–Jordan elimination .  9 1.5 Sums and scalar multiplications of matrices . 14 1.6 Products of vectors; Products of matrices . 18 1.7 Inverse matrices; One-sided and two-sided inverses . 24 1.8 Elementary matrices and forward elimination  . 26 1.9 When and how can one nd the inverse of a matrix? . 31 1.10 Applications .  36 1.10.1 Sums and products of block matrices  . 36 1.10.2 LU decompositions  . 38 1.10.3 Matrix technique for cryptography. 45 1.10.4 Systems of linear equations in electrical networks. 47 1.10.5 Leontief input-output model . 48 1.11 Exercises  . 52 2 Determinants; The signed area and the signed volume 59 2.1 Leibniz's letter to de l'Hopital . 59 2.2 Areas of parallelograms and determinants . 60 2.3 Denition and properties of the determinant . 63 2.4 Existence and uniqueness of the determinant .  68 2.5 Minors and cofactor expansions .  74 2.6 Cramer's rule for solving a system of linear equations . 80 2.7 Applications . 83 2.7.1 Determinants of three well known matrices  . 83 2.7.2 Determinants of block matrices  . 88 2.7.3 Cauchy–Binet formula .  90 2.7.4 Sylvester's determinant identity . 92 2.7.5 Areas and volumes by determinants . 94 2.8 Exercises  . 100 3 Vector Spaces; Abstraction of the 3-dimensional space R3.  107 3.1 From the 3-space R3 to vector spaces  . 108 3.2 New vector spaces from old ones  . 112 3.2.1 Which subsets can be vector spaces?  . 112 3.2.2 Sums and intersections of subspaces  . 114 3.2.3 Subspace spanned by a set . 116 3.3 Linear dependence and independence . 119 3.4 Bases and coordinate systems . 122 3.5 Dimensions of vector spaces . 125 3.6 How to nd a basis? . 127 3.7 Row, column and null spaces . 131 3.8 Rank-Nullity theorem . 137 3.9 Applications  . 141 3.9.1 One-sided or two-sided invertibility . 141 3.9.2 Curve tting with correct data . 145 3.9.3 The Wronskian; Linear independence of functions  . 148 3.9.4 Rank decompositions . 150 3.10 Exercises  . 151 4 Linear Transformations; Operation preserving maps 157 4.1 Examples and properties of linear transformations  . 157 4.2 Constructing linear transformations  . 162 4.3 Invertible linear transformations  . 164 4.4 Coordinates and coordinate isomorphisms  . 168 4.5 Matrix representations of linear transformations  . 169 4.6 Change of bases; Change of coordinates  . 173 4.7 Similarity of matrices  . 177 4.8 Applications  . 182 4.8.1 Vector spaces of linear transformations  . 182 4.8.2 Dual spaces and adjoints  . 185 4.8.3 Homogeneous coordinate systems and graphics  . 190 4.9 Exercises  . 196 5 Inner Product Spaces; Abstraction of Euclidean geometry 203 5.1 Inner products; Abstraction of the dot product . 203 5.2 Geometry of vectors; Lengths, angles, orthogonality . 207 5.3 Matrix representations of inner products  . 211 5.4 Gram–Schmidt orthogonalization for rectangular coordinates . 213 5.5 Projections and their algebraic characterization . 219 5.6 Orthogonal projections and oblique projections. 221 5.7 Projection matrices; Matrix representations of projections  . 224 5.8 Orthogonal complements of the row and column spaces . 227 5.9 Orthogonal matrices are orthogonal transformations  . 230 5.10 Applications  . 234 5.10.1 Least squares solutions with normal equations . 234 5.10.2 Least squares solutions when the coefcient matrix is of full rank . 238 5.10.3 Curve tting with over-determined data  . 241 5.10.4 Projection matrices, again  . 244 5.10.5 QR or LQ decompositions of matrices of full rank . 248 5.11 Exercises  . 252 6 Diagonalization and powers of matrices 259 6.1 Eigenvalues and eigenvectors . 260 6.2 Properties of eigenvalues and eigenvectors  . 264 6.3 Diagonalization and spectral decompositions  . 267 6.4 Applications . 273 6.4.1 Fibonacci sequence . 273 6.4.2 Linear recurrence relations; LRR . 275 6.4.3 Linear difference equations; LdE . 280 6.4.4 Discrete dynamical systems . 285 6.4.5 Markov processes . 289 6.4.6 Linear differential equations; LDE  . 291 6.4.7 LDE $&%('*)+$ for a diagonalizable matrix ) . 295 6.5 Exponential matrices  . 299 6.6 Applications continued  . 303 6.6.1 Existence and uniqueness of solutions to an LDE . 303 6.6.2 Diagonalization of linear transformations  . 307 6.7 Exercises  . 309 7 Complex Vector Spaces; Abstraction of the complex  n-space Cn. 315 7.1 How different are complex vector spaces from real vector spaces? 316 7.2 Hermitian inner products .  320 7.3 Orthogonal projections, the complex case . 323 7.4 Hermitian, skew-Hermitian and unitary matrices . 326 7.5 Unitarily diagonalizable matrices . 331 7.6 Orthogonally diagonalizable matrices .336 7.7 Normal matrices . 338 7.8 Application .  341 7.8.1 The spectral decompositions of normal matrices . 341 7.8.2 Singular value decompositions; SVD . 345 7.8.3 How to nd a one-sided (left or right) inverse of )  . 356 7.8.4 Hilbert spaces; Inner product spaces for calculus . 361 7.9 Exercises  . 362 8 Jordan Canonical Forms; A block diagonalization 367 8.1 Jordan canonical forms and Jordan decompositions  . 368 8.2 How to nd the Jordan canonical form J . 373 8.3 Jordan blocks and chains of generalized eigenvectors . 378 8.4 How to nd a change of basis matrix Q. 382 8.5 /21 and 354 for a Jordan canonical matrix J . 387 8.6 Cayley–Hamilton theorem . 394 8.7 Computing )61by the Cayley–Hamilton theorem .  397 8.8 Applications  . 400 8.8.1 The minimal polynomial of a matrix  . 400 8.8.2 Computing Ak by using the minimal polynomial . 405 8.8.3 Computing ea by using the minimal polynomial . 409 8.8.4 LdE xn=Axn-1 for any square matrix A . 414 8.9 Exercises . 418 9 Quadratic and bilinear forms 425 9.1 Quadratic forms; q(x)=xh Ax . 425 9.2 Diagonalization of quadratic forms . 432 9.3 The inertia classies level surfaces .  436 9.4 Characterizations of definite forms . 440 9.5 Change of bases for quadratic forms and the congruence relation . 444 9.6 How to compute the inertia . 446 9.7 Bilinear and sesquilinear forms; b(x,y)=xh Ay . 449 9.8 Diagonalization of symmetric bilinear or Hermitian sesquilinear forms . 452 9.9 Applications . 457 9.9.1 Nondegenerate bilinear forms .  457 9.9.2 Sylvester's law of inertia .  458 9.9.3 Real quadratic forms to symmetric bilinear forms . 462 9.9.4 Minimax of real-valued functions on Rn. 464 9.9.5 Minimax of quadratic forms and nding eigenvalues . 469 9.10 Exercises .474 Selected Answers and Hints 481 Bibliography 521 Index 522

곽진호(郭振鎬, Jin Ho Kwak) 북경교통대학 수학과와 한능신능원학원 소속 중국정부 특별초빙 교수

 

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