Elementary Abstract Algebra- Examples and Applications, 2019a

CONTENTS 9
8.5 Sigma notation in linear algebra . . . . . . . . . . . . . . 315
8.5.1 Matrix multiplication . . . . . . . . . . . . . . . . . . 316
8.5.2 The identity matrix and the Kronecker delta . . . . . 319
8.5.3 Abbreviated matrix notations . . . . . . . . . . . . . . 324
8.5.4 Matrix transpose and matrix inverse . . . . . . . . . . 326
8.5.5 Rotation matrices (in 3 dimensions) . . . . . . . . . . 328
8.5.6 Matrix traces . . . . . . . . . . . . . . . . . . . . . . . 331
8.6 Levi-Civita symbols and applications . . . . . . . . . . . . . . 334
8.6.1 Levi-Civita symbols: definitions and examples . . . . . 334
8.6.2 Levi-Civita symbols and determinants . . . . . . . . . 336
8.6.3 Levi-Civita symbols and cross products . . . . . . . . 342
8.6.4 Proof of the BAC-CAB Rule . . . . . . . . . . . . . . 344
8.6.5 Proof of Euler’s Rotation Theorem . . . . . . . . . . . 348
8.7 Summation by parts . . . . . . . . . . . . . . . . . . . . . . . 354
8.8 Hints for “Sigma Notation” exercises . . . . . . . . . . . . . . 360
8.9 Study guide for “Sigma Notation” chapter . . . . . . . . . . . 362
9 Polynomials 366
9.1 Why study polynomials? . . . . . . . . . . . . . . . . . . . . . 366
9.2 Review of polynomial arithmetic . . . . . . . . . . . . . . 369
9.3 Polynomial operations in summation notation . . . . . . . . . 371
9.4 More exotic polynomials . . . . . . . . . . . . . . . . . . . . . 378
9.5 Polynomial properties and summation notation . . . . . . . . 384
9.6 Polynomials and division . . . . . . . . . . . . . . . . . . . . . 392
9.6.1 The Division Algorithm for polynomials over fields 392
9.6.2 Greatest common divisors of polynomials . . . . . . . 396
9.6.3 Polynomial roots and the FTOA (easy part) . . 399
9.6.4 Algebraic closure and the FTOA (hard part) . . . . . 406
9.7 Hints for “Polynomial Rings” exercises . . . . . . . . . . . . . 411