## Applied Linear Algebra and Matrix AnalysisThis book is about matrix and linear algebra, and their applications. For many students the tools of matrix and linear algebra will be as fundamental in their professional work as the tools of calculus; thus it is important to ensure that students appreciate the utility and beauty of these subjects as well as the mechanics. To this end, applied mathematics and mathematical modeling ought to have an important role in an introductory treatment of linear algebra. In this way students see that concepts of matrix and linear algebra make concrete problems workable. In this book we weave signi?cant motivating examples into the fabric of the text. I hope that instructors will not omit this material; that would be a missed opportunity for linear algebra! The text has a strong orientation toward numerical computation and applied mathematics, which means that matrix analysis plays a central role. All three of the basic components of l- ear algebra — theory, computation, and applications — receive their due. The proper balance of these components gives students the tools they need as well as the motivation to acquire these tools. Another feature of this text is an emphasis on linear algebra as an experimental science; this emphasis is found in certain examples, computer exercises, and projects. Contemporary mathematical software make ideal “labs” for mathematical experimentation. Nonetheless, this text is independent of speci?c hardware and software pl- forms. Applications and ideas should take center stage, not software. |

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### Contents

LINEAR SYSTEMS OF EQUATIONS | 1 |

12 Notation and a Review of Numbers | 9 |

Basic Ideas | 21 |

General Procedure | 33 |

15 Computational Notes and Projects | 46 |

MATRIX ALGEBRA | 55 |

22 Matrix Multiplication | 62 |

23 Applications of Matrix Arithmetic | 71 |

43 Orthogonal and Unitary Matrices | 233 |

44 Change of Basis and Linear Operators | 242 |

45 Computational Notes and Projects | 247 |

THE EIGENVALUE PROBLEM | 251 |

52 Similarity and Diagonalization | 263 |

53 Applications to Discrete Dynamical Systems | 272 |

54 Orthogonal Diagonalization | 282 |

55 Schur Form and Applications | 287 |

24 Special Matrices and Transposes | 86 |

25 Matrix Inverses | 101 |

26 Basic Properties of Determinants | 114 |

27 Computational Notes and Projects | 129 |

VECTOR SPACES | 145 |

32 Subspaces | 161 |

33 Linear Combinations | 170 |

34 Subspaces Associated with Matrices and Operators | 183 |

35 Bases and Dimension | 191 |

36 Linear Systems Revisited | 198 |

37 Computational Notes and Projects | 208 |

GEOMETRICAL ASPECTS OF STANDARD SPACES | 211 |

42 Applications of Norms and Inner Products | 221 |

56 The Singular Value Decomposition | 291 |

57 Computational Notes and Projects | 294 |

GEOMETRICAL ASPECTS OF ABSTRACT SPACES | 305 |

62 Inner Product Spaces | 312 |

63 GramSchmidt Algorithm | 323 |

64 Linear Systems Revisited | 333 |

65 Operator Norms | 342 |

66 Computational Notes and Projects | 348 |

Table of Symbols | 355 |

Solutions to Selected Exercises | 357 |

375 | |

377 | |

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### Common terms and phrases

addition and scalar algorithm apply arithmetic augmented matrix calculate coefficient matrix complex numbers compute coordinates Corollary defined definition determinant diagonalizable eigenvalues eigenvectors elementary matrices elementary operation elementary row operations elements Example Exercise 11 Exercises and Problems fact following matrices formula function Gauss-Jordan elimination Gaussian elimination geometrical given graph Hermitian idea inner product space invertible invertible matrix laws leading entry linear algebra linear combination linear operator linear system linearly independent m x n matrix multiplication n x n matrix nonnegative nonzero norm notation null space obtain orthogonal orthogonal matrix orthonormal polynomial Problem 19 projection Proof rank real number reduced row echelon right-hand side row echelon form scalar multiplication Section set of vectors Show solution set solve spanning set square matrix standard inner product standard vector subset subspace Suppose symmetric system Ax system of equations Theorem transpose upper triangular vector space Verify vertex zero