A Modern Introduction to Linear Algebra

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ISBN: 9781439800409 | 1439800405
Cover: Hardcover
Copyright: 10/21/2009

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Taking a computational, algebraic, and geometric approach to linear algebra, A Modern Introduction to Linear Algebra provides an appealing foundation for a one-semester undergraduate course. This matrix-oriented text focuses on useful concepts at the heart of linear algebra, including finite-dimensional real vector spaces, eigenvalues, quadratic forms, and more. The author presents theoretical and applied examples as well as discussions of numerical aspects. In the appendices, he reviews set theory, sigma notation, mathematical induction, and complex numbers. A solutions manual is available for qualifying instructors.

Author	p. xi
Introduction	p. xiii
Vectors	p. 1
Vectors in Rⁿ	p. 1
Euclidean n-Space	p. 3
Vector Addition/Subtraction	p. 4
Scalar Multiplication	p. 6
Geometric Vectors in R² and R³	p. 7
Algebraic Properties	p. 8
The Inner Product and Norm	p. 12
The Angle between Vectors	p. 16
Spanning Sets	p. 27
Linear Independence	p. 36
Bases	p. 45
Coordinates/Change of Basis	p. 53
Subspaces	p. 58
Summary	p. 64
Systems of Equations	p. 67
The Geometry of Systems of Equations in R² and R³	p. 68
Matrices and Echelon Form	p. 77
The Matrix of Coefficients	p. 77
Elementary Row Operations	p. 79
Gaussian Elimination	p. 91
Computational Considerations-Pivoting	p. 97
Gauss-Jordan Elimination and Reduced Row Echelon Form	p. 104
Ill-Conditioned Systems of Linear Equations	p. 116
Rank and Nullity of a Matrix	p. 123
Row Spaces and Column Spaces	p. 123
The Null Space	p. 132
Systems of m Linear Equations in n Unknowns	p. 137
The Solution of Linear Systems	p. 138
Summary	p. 154
Matrix Algebra	p. 157
Addition and Subtraction of Matrices	p. 157
Scalar Multiplication	p. 160
Transpose of a Matrix	p. 161
Matrix-Vector Multiplication	p. 165
Matrix-Vector Multiplication as a Transformation	p. 172
The Product of Two Matrices	p. 182
The Column View of Multiplication	p. 185
The Row View of Multiplication	p. 188
Positive Powers of a Square Matrix	p. 193
Partitioned Matrices	p. 203
Inverses of Matrices	p. 210
Negative Powers of a Square Matrix	p. 217
Elementary Matrices	p. 220
The LU Factorization	p. 234
Summary	p. 249
Eigenvalues, Eigenvectors, and Diagonalization	p. 253
Determinants	p. 254
Cramer's Rule	p. 268
Determinants and Geometry	p. 277
The Manual Calculation of Determinants	p. 284
Eigenvalues and Eigenvectors	p. 293
Similar Matrices and Diagonalization	p. 311
Algebraic and Geometric Multiplicities of Eigenvalues	p. 321
The Diagonalization of Real Symmetric Matrices	p. 326
The Cayley-Hamilton Theorem (a First Look)/the Minimal Polynomial	p. 333
Summary	p. 344
Vector Spaces	p. 347
Vector Spaces	p. 347
Subspaces	p. 353
The Sum and Intersection of Subspaces	p. 357
Linear Independence and the Span	p. 365
Bases and Dimension	p. 373
Summary	p. 386
Linear Transformations	p. 389
Linear Transformations	p. 389
The Range and Null Space of a Linear Transformation	p. 399
The Algebra of Linear Transformations	p. 408
Matrix Representation of a Linear Transformation	p. 414
Invertible Linear Transformations	p. 425
Isomorphisms	p. 434
Similarity	p. 441
Similarity Invariants of Operators	p. 453
Summary	p. 457
Inner Product Spaces	p. 461
Complex Vector Spaces	p. 461
Inner Products	p. 468
Orthogonality and Orthonormal Bases	p. 477
The Gram-Schmidt Process	p. 483
Unitary Matrices and Orthogonal Matrices	p. 493
Schur Factorization and the Cayley-Hamilton Theorem	p. 505
The QR Factorization and Applications	p. 512
Orthogonal Complements	p. 520
Projections	p. 530
Summary	p. 535
Hermitian Matrices and Quadratic Forms	p. 539
Linear Functionals and the Adjoint of an Operator	p. 539
Hermitian Matrices	p. 547
Normal Matrices	p. 554
Quadratic Forms	p. 560
Singular Value Decomposition	p. 569
The Polar Decomposition	p. 579
Summary	p. 585
Basics of Set Theory	p. 587
Summation and Product Notation	p. 593
Mathematical Induction	p. 597
Complex Numbers	p. 601
Answers/Hints to Odd-Numbered Problems	p. 605
Index	p. 645
Table of Contents provided by Ingram. All Rights Reserved.

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A Modern Introduction to Linear Algebra

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Table of Contents

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