Page 15 - Advanced Linear Algebra

Contents xvii

When Is a Tensor Product Zero?, 367
Coordinate Matrices and Rank, 368
Characterizing Vectors in a Tensor Product, 371
Defining Linear Transformations on a Tensor Product, 374
The Tensor Product of Linear Transformations, 375
Change of Base Field, 379
Multilinear Maps and Iterated Tensor Products, 382
Tensor Spaces, 385
Special Multilinear Maps, 390
Graded Algebras, 392
The Symmetric and Antisymmetric
Tensor Algebras, 392
The Determinant, 403
Exercises, 406
15 Positive Solutions to Linear Systems:
Convexity and Separation, 411
Convex, Closed and Compact Sets, 413
Convex Hulls, 414
Linear and Affine Hyperplanes, 416
Separation, 418
Exercises, 423
16 Affine Geometry, 427
Affine Geometry, 427
Affine Combinations, 428
Affine Hulls, 430
The Lattice of Flats, 431
Affine Independence, 433
Affine Transformations, 435
Projective Geometry, 437
Exercises, 440
17 Singular Values and the Moore–Penrose Inverse, 443
Singular Values, 443
The Moore–Penrose Generalized Inverse, 446
Least Squares Approximation, 448
Exercises, 449
18 An Introduction to Algebras, 451
Motivation, 451
Associative Algebras, 451
Division Algebras, 462
Exercises, 469
19 The Umbral Calculus, 471
Formal Power Series, 471
The Umbral Algebra, 473