Lin I.H. — Geometric Linear Algebra. Vol. 1 :: Электронная библиотека попечительского совета мехмата МГУ

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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:

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

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Предметный указатель

Function, inverse      684
Function, invertible      684
Function, left inverse      683
Function, one-to-one      682
Function, onto      682
Function, restriction      683
Function, right inverse      683
Generalized eigenspace      210 224 479 514 515 516 794 801
Generalized eigenspace cycle (generated by a vector)      209 210 801
Generalized eigenspace order (of a vector)      801
Generalized eigenspace, A-cycle (generated by a vector)      210 570 802
Generalized eigenspace, annihilator, A-annihilator      213 214 570 802
Generalized eigenspace, cyclic invariant subspace      802
Generalized eigenspace, generalized eigenvector      514 515 516 794
Generalized eigenspace, minimal polynomial of a vector $\vec{x}$ related to $\mathrm{A}_{n \times m}$       802
Generalized eigenspace, root space      801
Generalized or pseudo-inverse (of a matrix)      176 419 429 642 767
Generalized or pseudo-inverse (of a matrix) column space      124 407 757
Generalized or pseudo-inverse (of a matrix) least square problem      419 429 765
Generalized or pseudo-inverse (of a matrix) left kernel      124 407 757
Generalized or pseudo-inverse (of a matrix) of $\mathrm{A}_{2 \times 3}$ and $\mathrm{A}_{n \times m}$ ( $m \leq n$ and r(A)=m)      419 456 462 763 765
Generalized or pseudo-inverse (of a matrix) of $\mathrm{A}_{2 \times 3}$ and $\mathrm{A}_{n \times m}$ ( $m \leq n$ and r(A)=m) right inverse      457 760
Generalized or pseudo-inverse (of a matrix) of $\mathrm{A}_{2 \times 3}$ and $\mathrm{A}_{n \times m}$ ( $m \leq n$ and r(A)=m) right invertible      457 760
Generalized or pseudo-inverse (of a matrix) of $\mathrm{A}_{3 \times 2}$ (with $\mathrm{r}(\mathrm{A})=\mathrm{r} \leq \textrm{min}(\mathrm{m},\mathrm{n})$ )      419 429 767 768
Generalized or pseudo-inverse (of a matrix) of $\mathrm{A}_{3 \times 2}$ and $\mathrm{A}_{n \times m}$ ( $m \geq n$ and r(A)=m)      429 462 469 763 765
Generalized or pseudo-inverse (of a matrix) of $\mathrm{A}_{3 \times 2}$ and $\mathrm{A}_{n \times m}$ ( $m \geq n$ and r(A)=m) left inverse      464 760
Generalized or pseudo-inverse (of a matrix) of $\mathrm{A}_{3 \times 2}$ and $\mathrm{A}_{n \times m}$ ( $m \geq n$ and r(A)=m) left invertible      463 760
Generalized or pseudo-inverse (of a matrix) optimal solution      419 429 767
Generalized or pseudo-inverse (of a matrix) right kernel      124 407 757
Generalized or pseudo-inverse (of a matrix) row space (of a matrix $\mathrm{A}_{n \times m}$ )      124 407 757
Geometric interpretation of a determinant of order      3 638
Gersgoerin's disk theorem (for eigenvalues)      490
Gram — Schmidt orthogonalization process      776
Group      686
Group, abelian, commutative      687
Group, affine group or group of affine motions on $\mathbb{R}^\mathrm{n}$       238 240 580 581 656
Group, general linear group on $\mathbb{F}^\mathrm{n}$       737
Group, group isomorphism      120
Group, projective group or group of projective transformations on $\mathrm{P}^\mathrm{n} (\mathbb{R})$ n=2, 3      660
Group, real general linear group on $\mathbb{R}^\mathrm{n}$       120 406 737
Group, subgroup      245 687
Half-space      652
Half-space half line or ray      653
Half-space segment      653
Half-space side (of a point w. r. t. a subspace)      653
Half-space, closed      653
Half-space, open      653
Homogeneous coordinate (of a point w. r. t. a basis)      306 311
Homogeneous linear equation      87 124 377 415 424 723
Homogeneous linear equation, general theory      723
Homogeneous linear equation, general theory, consistent      723
Homogeneous linear equation, general theory, fundamental solution      723
Homogeneous linear equation, general theory, general solution      723
Homogeneous linear equation, general theory, solution space      724
Hyperbola      303 304 305
Hyperbolic rotation      313
Hyperbolic rotation, transitive one-parameter subgroup of $\mathrm{G}_a (2;\mathbb{R})$       313
Hyperboloid      670
Hyperboloid of one-sheet      456 670
Hyperboloid of two-sheets      670
Hyperboloid, hyperbolic cylinder      671
Hyperboloid, hyperbolic paraboloid      671
Hyperplane      378
Hyperplane at infinity      657
Hypersubspace      378 751
Idempotent (linear) operator, matrix      111 707 740
Identitical vectors      7 25
Identity matrix      701
Indefinite matrix      789
Induced quotient operator      211
Infinite line      315 657
Infinite or ideal point      76 315 657
Infinite-dimensional vector space      697
Inhomogeneous or nonhomogeneous linear equation      88 150 377 415 424 442 724 759
Inhomogeneous or nonhomogeneous linear equation connection with generalized inverse of a matrix      765
Inhomogeneous or nonhomogeneous linear equation general theory      724 759
Inhomogeneous or nonhomogeneous linear equation general theory, augmented matrix      724
Inhomogeneous or nonhomogeneous linear equation general theory, particular solution      724
Inhomogeneous or nonhomogeneous linear equation general theory, solution affine subspace      724
Inhomogeneous or nonhomogeneous linear equation, three equations in two unknowns      415
Inhomogeneous or nonhomogeneous linear equation, three equations in two unknowns, augmented matrix      416
Inhomogeneous or nonhomogeneous linear equation, three equations in two unknowns, coefficient matrix      416
Inhomogeneous or nonhomogeneous linear equation, three equations in two unknowns, distance      418
Inhomogeneous or nonhomogeneous linear equation, three equations in two unknowns, generalized or pseudo-inverse      419
Inhomogeneous or nonhomogeneous linear equation, three equations in two unknowns, orthogonal projection      418 420
Inhomogeneous or nonhomogeneous linear equation, three equations in two unknowns, solution affine subspace      417
Inhomogeneous or nonhomogeneous linear equation, three equations in two unknowns, solution space      417
Inhomogeneous or nonhomogeneous linear equation, two equation in three unknowns      424
Inhomogeneous or nonhomogeneous linear equation, two equation in three unknowns, augmented matrix      425
Inhomogeneous or nonhomogeneous linear equation, two equation in three unknowns, coefficient matrix      425
Inhomogeneous or nonhomogeneous linear equation, two equation in three unknowns, distance      426
Inhomogeneous or nonhomogeneous linear equation, two equation in three unknowns, generalized or pseudo-inverse      429
Inhomogeneous or nonhomogeneous linear equation, two equation in three unknowns, orthogonal projection      428 429
Inhomogeneous or nonhomogeneous linear equation, two equation in three unknowns, particular solution      425 426
Inhomogeneous or nonhomogeneous linear equation, two equation in three unknowns, solution affine subspace      425 426
Inhomogeneous or nonhomogeneous linear equation, two equation in three unknowns, solution space      425 426
Inner product space      169 773
Inner product space, adjoint (or conjugate) operator of a linear operator      781
Inner product space, adjoint (or conjugate) operator of a linear operator, Hermitian or self-adjoint operator (matrix)      782
Inner product space, adjoint (or conjugate) operator of a linear operator, normal operator (matrix)      782
Inner product space, adjoint (or conjugate) operator of a linear operator, orthogonal operator (matrix)      782
Inner product space, adjoint (or conjugate) operator of a linear operator, skew-Hermitian operator (matrix)      782
Inner product space, adjoint (or conjugate) operator of a linear operator, skew-symmetric operator (matrix)      782
Inner product space, adjoint (or conjugate) operator of a linear operator, symmetric (or self-adjoint) operator (matrix)      782
Inner product space, adjoint (or conjugate) operator of a linear operator, unitary operator (matrix)      782
Inner product space, angle (between two nonzero vectors)      774
Inner product space, Cauchy — Schwarz inequality      774
Inner product space, complex inner product space $(\mathrm{V}, \langle , \rangle)$ or V      773
Inner product space, complex inner product space $(\mathrm{V}, \langle , \rangle)$ or V, unitary space      773
Inner product space, Gram — Schmidt orthogonalization process      776
Inner product space, inner product, (abstract) general definition      773
Inner product space, inner product, bilinear      774
Inner product space, inner product, conjugate symmetric (complex), symmetric      773
Inner product space, inner product, linear      773
Inner product space, inner product, positive definite      773
Inner product space, matrix representation of an inner product      776
Inner product space, matrix representation of an inner product, congruent      776
Inner product space, matrix representation of an inner product, Hermitian (positive-definite)      776
Inner product space, matrix representation of an inner product, symmetric (positive-definite)      776
Inner product space, norm or length of a vector      360 774
Inner product space, orthogonal complement (of a set)      778
Inner product space, orthogonal decomposition (of an inner product space)      778
Inner product space, orthogonal decomposition (of an inner product space), Bessel inequality      779
Inner product space, orthogonal decomposition (of an inner product space), optimal approximation inequality      779
Inner product space, orthogonal decomposition (of an inner product space), orthogonal projection vector      778
Inner product space, orthogonal decomposition (of an inner product space), orthogonal vector      778
Inner product space, orthogonal decomposition (of an inner product space), Pythagorean theorem      778
Inner product space, orthogonal or perpendicular (of two sets of vectors)      778
Inner product space, orthogonal or perpendicular (of two vectors)      774
Inner product space, orthogonal projection (of a vector on another)      775
Inner product space, orthogonal projection (operator)      779
Inner product space, orthogonal set or basis      775
Inner product space, orthonormal basis      775
Inner product space, orthonormal basis, standard or natural orthonormal basis for $\mathbb{R}^\mathrm{n}$ or $\mathbb{C}^\mathrm{n}$       775
Inner product space, real inner product space $(\mathrm{V}, \langle , \rangle)$ or V      773
Inner product space, Riesz representation theorem      780
Inner product space, Riesz representation theorem, conjugate linear      781
Inner product space, Riesz representation theorem, self-dual inner product space      781
Inner product space, signed orthogonal projection (as a quantity)      774
Inner product space, triangle inequality      774
Invariance under similarity determinant (of a linear operator)      121 407
Invariance under similarity determinant (of a linear operator) characteristic polynomial      121 407
Invariance under similarity determinant (of a linear operator) eigenvalue      121 407
Invariance under similarity determinant (of a linear operator) rank      121 407
Invariance under similarity determinant (of a linear operator) trace      121 407
Invariant subspace      85 188 193 210 591 748
Invariant subspace, line      188 591
Invariant subspace, point      90 591
Invariant, affine      19 90 286 374 636 638 679
Invariant, projective      300 317 661

Inverse matrix      47 55 336 338 711 731
Inverse vector      9 13 26 692
Invertible (linear) operator or transformation      16 35 47 84 336 338 732
Invertible matrix      47 55 336 338 711 731
Invertible matrix, left invertible (inverse)      142 423 760
Invertible matrix, right invertible (inverse)      142 423 760
Involutory operator or matrix      112 544 708 741
Involutory operator or matrix, reflection      544
Irreducible quadratic curve      306
Jordan canonical form      222 224 516 541 549 799 805 806
Jordan canonical form, exponential of      521 534 546
Jordan canonical form, Jordan block      520 525 546 800 806
Jordan canonical form, Jordan canonical basis      222 224 516 528 529 549 806
Kernel (of a linear transformation)      56 85 124 366 383 407 734
Kernel (of a linear transformation), left kernel      124 407 757
Kernel (of a linear transformation), right kernel      124 407 757
Lagrange interpolation theorem (formula)      441 699
Lagrange interpolation theorem (formula), polynomial      441 699
Laplace expansion (of determinant)      730
LDL*-decomposition (of matrix)      162 442
LDU-decomposition (of matrix)      162 442
Left inverse (of a matrix)      142 423 464 761
Left invertible matrix      142 423 463 760
Left kernel (or nullspace)      124 407 757
Length of a vector      360 774
Lie algebra      146
Lie algebra, bilinearity      146
Lie algebra, Jacobi identity      146
Lie algebra, Lie product or bracket operation (of two linear operators)      146
Lie algebra, product vector (of two vectors)      146
Limit matrix      451
Line in affine or vector space $\mathbb{R}$ : coordinate representation w. r. t. an affine basis      18
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , coordinate equation w. r. t. a basis      61 346
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , coordinate equations of coordinate axes in an affine basis (coordinate system)      60 346 363
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , parametric equation in vector direction      61 346
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , parametric equation in vector form      61 346
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , parametric equation in vector point      61 346
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , parametric equation w. r. t. a basis      61 346
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , relative positions of a line and a plane      354
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , relative positions of a line and a plane, coincident      354
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , relative positions of a line and a plane, intersecting      354
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , relative positions of a line and a plane, parallel      354
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , relative positions of two lines      63 349
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , relative positions of two lines, coincident      63 349
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , relative positions of two lines, intersecting      63 349
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , relative positions of two lines, parallel      63 349
Line in affine or vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ , relative positions of two lines, skew (in $\mathbb{R}^3$ )      349
Linear combination (of vectors) in $\mathbb{R}^1$       8 9
Linear combination (of vectors) in $\mathbb{R}^2$       31
Linear combination (of vectors) in $\mathbb{R}^3$       324
Linear combination (of vectors) in general vector space      695
Linear combination (of vectors), coefficients (scalars)      8 9 31 324 695
Linear dependence (of vectors) in $\mathbb{R}^1$       9
Linear dependence (of vectors) in $\mathbb{R}^2$       32 33 40
Linear dependence (of vectors) in $\mathbb{R}^3$       324
Linear dependence (of vectors) in general vector space      696
Linear equation      723 724
Linear equation fundamental solution      723
Linear equation general solution      723
Linear equation particular solution      724
Linear equation solution affine subspace      724
Linear equation solution space      723
Linear equation, consistent equation      723
Linear functional general theory in vector space over a field      84 378 420 732 749
Linear functional general theory in vector space over a field, adjoint or dual operator      422 754
Linear functional general theory in vector space over a field, annihilator      421 752
Linear functional general theory in vector space over a field, dual basis      420 750
Linear functional general theory in vector space over a field, dual space      420 749
Linear functional general theory in vector space over a field, hyperplane (or hypersubspace)      378 751
Linear functional general theory in vector space over a field, Lagrange multiplier's method      749
Linear functional general theory in vector space over a field, natural linear isomorphism      421 752
Linear functional general theory in vector space over a field, second dual basis      421 751
Linear functional general theory in vector space over a field, symmetric bilinear      756
Linear functional in inner product space      780
Linear functional in inner product space, adjoint operator      781
Linear functional in inner product space, Riesz representation theorem      780
Linear independence (of vectors) in $\mathbb{R}^1$       10
Linear independence (of vectors) in $\mathbb{R}^2$       33 41
Linear independence (of vectors) in $\mathbb{R}^3$       325 336
Linear independence (of vectors) in general vector space      696
Linear isomorphism      11 18 35 58 84 117 136 326 373 732 734
Linear isomorphism, natural      421 752
Linear operator      84 89 366 406 435 732 781
Linear operator geometric mapping properties      90 286 374
Linear operator in general vector space, adjoint or dual operator      422 754
Linear operator in general vector space, differential operator      440
Linear operator in general vector space, idempotent operator      110 707 740
Linear operator in general vector space, integral operator      440
Linear operator in general vector space, involutory operator      112 544 708 741
Linear operator in general vector space, iterative linear operator      120 141 438 737
Linear operator in general vector space, nilpotent operator      110 215 537 539 708 803
Linear operator in general vector space, nilpotent operator invariant system      540 541
Linear operator in general vector space, nilpotent operator Jordan canonical form      539 542 803
Linear operator in general vector space, nilpotent operator power, index      215 537 539 708 803
Linear operator in general vector space, nilpotent operator square root      543
Linear operator in inner product space      781
Linear operator in inner product space, adjoint operator      781
Linear operator in inner product space, Hermitian operator (matrix)      782
Linear operator in inner product space, normal operator (matrix)      782
Linear operator in inner product space, orthogonal operator (matrix)      782
Linear operator in inner product space, skew-Hermitian operator (matrix)      782
Linear operator in inner product space, skew-symmetric operator (matrix)      782
Linear operator in inner product space, symmetric or self-adjoint operator (matrix)      782
Linear operator in inner product space, unitary operator (matrix)      782
Linear operator invariant, characteristic polynomial      121 407
Linear operator invariant, determinant      121 407
Linear operator invariant, eigenvalue      121 407
Linear operator invariant, rank      124 407
Linear operator invariant, trace      122 407
Linear operator matrix and its transpose      756
Linear operator, canonical form, diagonal      193 198 385 479 796
Linear operator, canonical form, Jordan      224 393 394 395 399 516 799 805
Linear operator, canonical form, rational      233 390 398 399 562 809 812
Linear operator, classification of linear operators on $\mathbb{R}^2$ , $\mathbb{R}^3$ , enlargement      95 385
Linear operator, classification of linear operators on $\mathbb{R}^2$ , $\mathbb{R}^3$ , one-way stretch      385
Linear operator, classification of linear operators on $\mathbb{R}^2$ , $\mathbb{R}^3$ , reflection      96 387
Linear operator, classification of linear operators on $\mathbb{R}^2$ , $\mathbb{R}^3$ , shearing      100 392
Linear operator, classification of linear operators on $\mathbb{R}^2$ , $\mathbb{R}^3$ , three-way stretch      385
Linear operator, classification of linear operators on $\mathbb{R}^2$ , $\mathbb{R}^3$ , two-way stretch      95 385
Linear operator, invertible      84 89 117 144;
Linear or vector subspace, simply called subspace      40 85 330 694 695
Linear or vector subspace, simply called subspace, direct sum subspace      41 330 694
Linear or vector subspace, simply called subspace, generated or spanned by a set of vectors      41 330 695
Linear or vector subspace, simply called subspace, intersection subspace      330 694
Linear or vector subspace, simply called subspace, invariant subspace      85 748
Linear or vector subspace, simply called subspace, invariant subspace, (whole) space      85
Linear or vector subspace, simply called subspace, invariant subspace, image or range subspace      85
Linear or vector subspace, simply called subspace, invariant subspace, kernel or null subspace      85
Linear or vector subspace, simply called subspace, invariant subspace, zero subspace      85
Linear or vector subspace, simply called subspace, proper      41
Linear or vector subspace, simply called subspace, sum subspace      41 330 694
Linear or vector subspace, simply called subspace, zero subspace      41 694
Linear space      see “Vector space”
Linear transformation (mapping)      11 16 18 35 47 56 57 58 84 87 116 136 326 338 343 366 406 435 732
Linear transformation (mapping) decomposable      749
Linear transformation (mapping) image or range subspace      56 85 343 366 734
Linear transformation (mapping) kernel or null subspace      56 85 343 366 734
Linear transformation (mapping) left invertible (inverse)      142
Linear transformation (mapping) linear isomorphism      11 16 35 47 84 326 338 373 732 734
Linear transformation (mapping) matrix representation w. r. t. bases      57 58 59 86 99 115 172 198 211 366 369 372 374 407 413 580 735
Linear transformation (mapping) matrix representation w. r. t. bases, change of coordinates matrix      47 337 735
Linear transformation (mapping) matrix representation w. r. t. bases, diagonalizable      117 193 198 407 794
Linear transformation (mapping) matrix representation w. r. t. bases, transition matrix      47 337 735
Linear transformation (mapping) normal form (of a linear transformation or matrix)      140 164 437 457
Linear transformation (mapping) nullity      89 366 734
Linear transformation (mapping) projection (operator transformation)      136 435
Linear transformation (mapping) projection (operator transformation), natural projection      211 747
Linear transformation (mapping) projection (operator transformation), projectionalization (of a linear transformation)      137 436 746
Linear transformation (mapping) rank      89 366 407 734
Linear transformation (mapping) rank theorem (of a linear transformation or matrix)      140 436

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