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Lin I.H. — Geometric Linear Algebra. Vol. 1
Lin I.H. — Geometric Linear Algebra. Vol. 1

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Название: Geometric Linear Algebra. Vol. 1

Автор: Lin I.H.

Аннотация:

Writing for freshmen and sophomores in college or advanced high school students, Lin (National Taiwan Normal U.) takes the view that recently geometric linear algebra has become too algebraic and tackles the subject from another angle, starting with the affine and linear structures of real vector spaces R1, R2 and R3, stepping through them as one-dimensional (including affine variants), two- dimensional (including linear transformations and affine transformations) and three dimensions, including lines and planes in space. In appendices Lin gives the background information necessary to follow the text and do the exercises, including an extended description of algebraic linear algebra.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 856

Добавлена в каталог: 04.07.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Adjoint linear operator      422 754 781
Adjoint linear operator (also called) dual operator in dual space      422 754
Adjoint linear operator, self-adjoint or symmetric operator      782 786
Adjoint matrix      336 731
Affine basis (or frame)      9 10 19 31 34 71 239 324 326 362 580 640
Affine basis, base point (origin)      9 10 19 31 34 71 239 324 326 362 580 640
Affine basis, coordinate vector      10 34 326 640
Affine basis, orthonormal affine basis      613 625
Affine coordinate system      19 34 38 324 326 328 362 640
Affine coordinate system, affine coordinate      19 72 239 363 641
Affine coordinate system, barycenter      19 72 363
Affine coordinate system, barycentric coordinate      19 72 363 642
Affine dependence of points      32 71 239 325 362 580 640
Affine geometry      292 640
Affine geometry invariant      19 90 286 374 636 638 679
Affine group      238 240 580 581 656
Affine independence of points      33 71 239 325 362 580 640
Affine invariant      19 90 286 310 374 636
Affine space      9 12 31 34 235 324 326 640
Affine space, n-dimensional      235
Affine space, n-dimensional affine motion      237 240
Affine space, n-dimensional affine subspace      236
Affine space, n-dimensional affine transformation or mapping      237
Affine space, n-dimensional base point      235
Affine space, n-dimensional difference space      235
Affine space, n-dimensional difference vector      236
Affine space, n-dimensional free vector      235
Affine space, n-dimensional position vector (with initial point and terminal point)      235
Affine space, n-dimensional zero vector      235 236
Affine space, one-dimensional      8 12
Affine space, three-dimensional      324 326
Affine space, two-dimensional      31 36
Affine subspace      67 236 359 425 640 724
Affine subspace, coincident      645
Affine subspace, dimension      235
Affine subspace, dimension (or intersection) theorem      644
Affine subspace, hyperplane      641
Affine subspace, k-dimensional plane      640
Affine subspace, operation of affine intersection      644
Affine subspace, operation of affine subspaces      643
Affine subspace, operation of affine sum      644
Affine subspace, parallel      645
Affine subspace, relative positions of affine subspaces      645
Affine subspace, skew      645
Affine transformation or mapping or motion      16 18 47 67 73 83 237 250 326 338 364 366 580 581 655
Affine transformation or mapping or motion, (three-way) stretch      611
Affine transformation or mapping or motion, (three-way) stretch, enlargement      611 656
Affine transformation or mapping or motion, (three-way) stretch, orthogonal stretch      611
Affine transformation or mapping or motion, (three-way) stretch, similarity      611 656
Affine transformation or mapping or motion, decomposition of affine transformation      241 580
Affine transformation or mapping or motion, invariant affine subspace      251 591
Affine transformation or mapping or motion, invariant point      251 591
Affine transformation or mapping or motion, invariant point, subspace of invariant points      251 591
Affine transformation or mapping or motion, matrix representation      239 580 581
Affine transformation or mapping or motion, one-way stretch, stretching      255 596 597
Affine transformation or mapping or motion, one-way stretch, stretching, orthogonal one-way stretch      256 597
Affine transformation or mapping or motion, orthogonal projection      460 462 468 624 627 635
Affine transformation or mapping or motion, orthogonal projection, affinity      285
Affine transformation or mapping or motion, orthogonal projection, elliptic (orthogonal) rotation      283
Affine transformation or mapping or motion, orthogonal projection, hyperbolic (orthogonal) rotation      283
Affine transformation or mapping or motion, orthogonal reflection      274 461 624
Affine transformation or mapping or motion, reflection (skew)      251 591
Affine transformation or mapping or motion, reflection (skew), orthogonal reflection      253 274 592 624
Affine transformation or mapping or motion, reflection (skew), symmetric motion      253 592
Affine transformation or mapping or motion, regular or proper or nonsingular      251 655
Affine transformation or mapping or motion, rotation      271 272 620
Affine transformation or mapping or motion, shearing      265 266 611 612 613
Affine transformation or mapping or motion, shearing, skew shearing      282 635
Affine transformation or mapping or motion, singular      251
Affine transformation or mapping or motion, the fundamental theorem      241 580
Affine transformation or mapping or motion, translation      67 247 251 591
Affine transformation or mapping or motion, two-way stretch      261 604
Affine transformation or mapping or motion, two-way stretch, enlargement (scale factor)      262 605
Affine transformation or mapping or motion, two-way stretch, orthogonal two-way stretch      261
Affine, natural affine basis for $\mathbb{R}^2$      239
Affine, natural affine basis for $\mathbb{R}^3$      581
Algebra homomorphism      744
Algebra isomorphism      119 406
Algebraic rank (of a matrix)      125 367 383 407 739
Annihilator      213 214 421 752 757 802
Area coordinate      76
Area coordinate, affine coordinate system      76
Area coordinate, affine coordinate system base point      76
Area coordinate, affine coordinate system base triangle      76
Area coordinate, affine coordinate system vertex      76
Area coordinate, barycentric coordinate      76
Area coordinate, base point      76
Area coordinate, coordinate or base triangle      76
Area coordinate, homogeneous area coordinate      76
Area coordinate, ideal or infinite point      76
Area coordinate, normalized barycentric coordinate      76
Area coordinate, oriented triangle      75
Area coordinate, oriented triangle, signed area      75
Area coordinate, orthogonal or Descartesian or rectangular coordinate system      77
Area coordinate, signed distance      78
Augmented matrix      150 152 156 180 377 416 425 724
Barycentric coordinate for $\mathbb{R}$ (a line)      19
Barycentric coordinate for $\mathbb{R}$ (a line)barycenter (middle point of base segment)      19
Barycentric coordinate for $\mathbb{R}^2$ (a plane)      72
Barycentric coordinate for $\mathbb{R}^2$ (a plane), affine coordinate      72
Barycentric coordinate for $\mathbb{R}^2$ (a plane), barycenter (of base triangle)      72
Barycentric coordinate for $\mathbb{R}^2$ (a plane), base triangle      72
Barycentric coordinate for $\mathbb{R}^3$ (a space)      363 642
Barycentric coordinate for $\mathbb{R}^3$ (a space), affine coordinate      363
Barycentric coordinate for $\mathbb{R}^3$ (a space), barycenter (of base tetrahedron)      363 642
Barycentric coordinate for $\mathbb{R}^3$ (a space), base tetrahedron      363
Barycentric coordinate for $\mathbb{R}^3$ (a space), coordinate axis      363
Barycentric coordinate for $\mathbb{R}^3$ (a space), coordinate plane      363
Basis      9 10 31 34 71 324 326 330 336 697
Basis, basis vectors      9 10 31 34 71 324 326 697
Basis, changes of bases (coordinates)      16 46 117 337 406 735
Basis, natural or standard (orthonormal) basis for $\mathbb{R}^2$      38 55
Basis, natural or standard (orthonormal) basis for $\mathbb{R}^3$      328 335 366
Basis, natural or standard basis for $\mathbb{F}^\mathrm{n}$      693 697
Basis, ordered basis      9 10 34 324 326 331 698
Basis, Steinitz's replacement theorem      697
Bessel inequality      779
Bilinear form      304 756
Block or partitioned matrix      180
Block or partitioned matrix addition      181
Block or partitioned matrix conjugate transpose      181
Block or partitioned matrix Frobenius inequality (for ranks)      186
Block or partitioned matrix of the same type      181
Block or partitioned matrix product      181
Block or partitioned matrix rank decomposition theorem (of a matrix)      184
Block or partitioned matrix scalar multiplication      181
Block or partitioned matrix Schur's formula      182
Block or partitioned matrix Sylvester inequality (for ranks)      186
Block or partitioned matrix transpose      181
Block or partitioned matrix, pseudo-diagonal, lower or upper triangular      181
Block or partitioned matrix, submatrix      134 180
Canonical form (of a matrix)      559
Canonical form (of a matrix), canonical form of $3 \times 3$ real matrices under similarity      399
Canonical form (of a matrix), canonical or standard form of quadratic curves      303
Canonical form (of a matrix), canonical or standard form of quadrics: Euclidean, affine, projective      670 675 679
Canonical form (of a matrix), diagonal canonical form (of a square matrix)      193 198 386 399 479 718 797
Canonical form (of a matrix), Jordan canonical form (of a square matrix)      222 224 393 395 399 516 541 543 547—552 799 805
Canonical form (of a matrix), rational canonical form (of a square matrix)      233 390 398 399 560 562 574 809 812
Cauchy — Schwarz inequality      774
Cayley — Hamilton theorem      108 209 213 221 399 402 477 792
Ceva theorem      291 296 651
Change of coordinates      16 46 337
Change of coordinates formula      16 46 337
Change of coordinates, transition matrix or change of coordinates matrix      16 46 337 735
Characteristic equation      106 384 719 791
Characteristic of a field      686
Characteristic polynomial (of a linear operator or square matrix)      106 193 212 384 407 491 719 791 795
Characteristic root (eigenvalue)      105 384 407 719 790
Characteristic vector (eigenvector)      105 384 407 719 790
Co-dimension      747
Coefficient matrix      56 149 152 156 250 342 377 416 425 442
Column rank (of a matrix)      125 366 367 383 407
Column space      125 407 757
Column vector      125 367 375 703
Complete quadrilateral      299
Complete quadrilateral cross ratio      299 300 661
Complete quadrilateral harmonic points      299 300
Complete quadrilateral signed ratio      299
Congruence of matrix      166 454
Congruence of matrix, Sylvester's law of inertia (of symmetric bilinear form or symmetric matrix)      166 454
conic      300 306
Conic, center      306
Conic, central      306
Conic, degenerated      306
Conic, noncentral      306
Conic, nondegenerated      306
Conjugate (matrix) of a matrix      702
Conjugate (matrix) of a matrix, conjugate transpose      702 714
Convex set      653
Convex set, convex cell      653
Convex set, convex hull or closure      653
Coordinate system (also called affine basis)      9 10 31 34 46 324
Coordinate system (also called affine basis) (associated) vector space      9 10 31 34 324 326
Coordinate system (also called affine basis) affine or oblique coordinate system      38 328
Coordinate system (also called affine basis) basis (vector)      9 31 324
Coordinate system (also called affine basis) Cartesian or rectangular or natural coordinate system      38 328 335
Coordinate system (also called affine basis) origin      9 31 324
Coordinate system (also called affine basis) zero vector      9 31 324
Coordinate vector (of a point or a vector w. r. t. a basis) in $\mathbb{R}$      10
Coordinate vector (of a point or a vector w. r. t. a basis) in $\mathbb{R}^2$ (first, second component)      34
Coordinate vector (of a point or a vector w. r. t. a basis) in $\mathbb{R}^3$ (first, second, third component)      326
Coordinatized space (w. r. t. an affine basis) of a (three-dimensional) space      326
Coordinatized space (w. r. t. an affine basis) of a plane      35
Coordinatized space (w. r. t. an affine basis) of a straight line      11
Cross ratio (of four points)      300 661
Darboux theorem      291
Decomposition of matrices LU, LDU, LDL*, LPU, LS      161 162 167 442
Decomposition of matrices, lower and upper triangular decomposition (of a square matrix)      161
Decomposition of matrices, polar decomposition      772
Decomposition of matrices, rank decomposition theorem (of a matrix)      184
Decomposition of matrices, singular value decomposition theorem (SVD)      178 771
Desargues theorem      296
Determinant      727
Determinant adjoint matrix of a square matrix      731
Determinant cofactor      728 730
Determinant Cramer's rule      88 732
Determinant expansion along a row or a column      728
Determinant inverse matrix of an invertible matrix      731
Determinant Laplace expansion      730
Determinant minor      728
Determinant properties      728
Determinant, real determinant of order 2      47 55 106 121
Determinant, real determinant of order 3      336 337
Determinant, real determinant of order 3, adjoint matrix      336 731
Determinant, real determinant of order 3, cofactor      336
Determinant, real determinant of order 3, expansion along a row or a column      336
Determinant, real determinant of order 3, minor      336
Determinant, subdeterminant      730
Determinant, subdeterminant principal      730
Diagonalizable linear operator or matrix      117 193 198 212 385 399 479 718 794
Diagonalizable linear operator or matrix, characteristic polynomial (equation)      106 121 193 198 216 407 437 719 791 795
Diagonalizable linear operator or matrix, criteria for diagonalizability      198 212 479 718 796
Diagonalizable linear operator or matrix, criteria for diagonalizability, algebraic multiplicity      199 479 797
Diagonalizable linear operator or matrix, criteria for diagonalizability, geometric multiplicity      199 479 797
Diagonalizable linear operator or matrix, diagonal canonical form or decomposition (of a square matrix or linear operator)      199 479 797
Diagonalizable linear operator or matrix, eigenspace      193 198 478 791
Diagonalizable linear operator or matrix, eigenvalue or characteristic value (root)      105 106 190 193 198 385 477 719 790
Diagonalizable linear operator or matrix, eigenvector or characteristic vector      105 106 121 190 193 198 385 477 719 790
Diagonalizable linear operator or matrix, generalized eigenspace      210 224 516 794 801
Diagonalizable linear operator or matrix, generalized eigenvector      224 516 794
Diagonalizable linear operator or matrix, minimal polynomial      199 209 214 479 794 795
Diagonalizable linear operator or matrix, simultaneously diagonalizable      209 213
Differential equation      505
Differential equation, homogeneous      505
Differential equation, linear ODE with constant coefficients      509
Differential equation, nonhomogeneous      508 552
Dimension (of a vector space)      12 36 43 327 330 698
Dimension (of a vector space), dimension theorem      88 366 734 738
Dimension (of a vector space), finite-dimensional      697
Dimension (of a vector space), infinite-dimensional      697
Dimension (of a vector space), n-dimensional      698
Direct sum of restriction operators      210 211
Direct sum of subspaces      33 41 330 694 746
Directed (line) segment      6 7 65 356
Directed (line) segment coordinate or parametric representation      18 61 62 356
Directed (line) segment end point      18 65 356
Directed (line) segment exterior point      18 65 356
Directed (line) segment initial point      18 65 356
Directed (line) segment interior point      18 65 356
Directed (line) segment length      6
Directed (line) segment middle point      19 65 356
Directed (line) segment negative direction      6
Directed (line) segment positive direction      6
Directed (line) segment signed length      6
Directed (line) segment terminal point      18 65 356
Discriminant of a quadratic curve      306
Discriminant of a quadratic equation      689
Discriminant of a quadric      679
Dual basis      420 750
Dual space      420 749
Dual space, second      421 751
Echelon matrix      164 443 721
Echelon matrix, leading entry of a row (of a matrix)      721
Echelon matrix, pivot      162 164 721
Echelon matrix, pivot, free variable      721
Echelon matrix, pivot, pivot column (or row)      164 721
Echelon matrix, pivot, pivot matrix      162
Echelon matrix, pivot, pivot variable      721
Echelon matrix, row-reduced echelon matrix      164 443 721
Eigenspace      193 210 224 478 479 516 791 794
Eigenvalue (characteristic value, root)      105 190 193 224 384 477 479 516 719 794
Eigenvector (characteristic vector)      105 190 193 224 384 477 479 516 719 791
Elementary column or row operation      149 150 442 719
Elementary column or row operation, type 1      149 150 442 719 720
Elementary column or row operation, type 2      149 150 442 719 720
Elementary column or row operation, type 3      149 150 442 719 720
Elementary divisor      804
Elementary matrix      150 442 720
Elementary matrix, elementary matrix factorization (of an invertible matrix)      161 444 726
Elementary symmetric function (of n variables)      491
Elimination method      148 442 719
ellipse      303 304 305
Ellipse, imaginary ellipse      303 305
Ellipsoid      670
Ellipsoid, elliptic cone      670
Ellipsoid, elliptic cylinder      671
Ellipsoid, elliptic paraboloid      670
Ellipsoid, imaginary ellipsoid      670
Ellipsoid, imaginary elliptic cylinder      671
Ellipsoid, point ellipsoid      670
Elliptic rotation      312 313
Elliptic rotation, transitive one-parameter subgroup of $\mathrm{G}_a (2;\mathbb{R})$      313
Enlargement: scale, invariant point      95 247 262 385 611
f-cyclic subspace generated by a vector $\vec{x}$      209 210 802
f-invariant subspace      210
Field      684
Field characteristic      686
Field, complex ($\mathbb{C}$)      685
Field, finite field ($\mathbb{I}_\mathrm{p}$)      686
Field, infinite field      686
Field, rational ($\mathbb{Q}$)      686
Field, real ($\mathbb{R}$)      685
Finite-dimensional vector space      697
Free variable      721
Function      682
Function composite      683
Function identity      683
Function, domain, range      682
Function, equal      682
Function, image, preimage      682
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