## Linear Algebra"Linear Algebra" is intended for a one-term course at the junior or senior level. It begins with an exposition of the basic theory of vector spaces and proceeds to explain the fundamental structure theorem for linear maps, including eigenvectors and eigenvalues, quadratic and hermitian forms, diagnolization of symmetric, hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. The book also includes a useful chapter on convex sets and the finite-dimensional Krein-Milman theorem. The presentation is aimed at the student who has already had some exposure to the elementary theory of matrices, determinants and linear maps. However the book is logically self-contained. In this new edition, many parts of the book have been rewritten and reorganized, and new exercises have been added. |

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### Contents

Vector Spaces | 1 |

1 DEFINITIONS | 2 |

2 BASES | 10 |

3 DIMENSION OF A VECTOR SPACE | 15 |

4 SUMS AND DIRECT SUMS | 19 |

Matrices | 23 |

2 LINEAR EQUATIONS | 29 |

3 MULTIPLICATION OF MATRICES | 31 |

8 INVERSE OF A MATRIX | 174 |

9 THE RANK OF A MATRIX AND SUBDETERMINANTS | 177 |

Symmetric Hermitian and Unitary Operators | 180 |

2 HERMITIAN OPERATORS | 184 |

3 UNITARY OPERATORS | 188 |

Eigenvectors and Eigenvalues | 194 |

2 THE CHARACTERISTIC POLYNOMIAL | 200 |

3 EIGENVALUES AND EIGENVECTORS OF SYMMETRIC MATRICES | 213 |

Linear Mappings | 43 |

2 LINEAR MAPPINGS | 51 |

3 THE KERNEL AND IMAGE OF A LINEAR MAP | 59 |

4 COMPOSITION AND INVERSE OF LINEAR MAPPINGS | 66 |

5 GEOMETRIC APPLICATIONS | 72 |

Linear Maps and Matrices | 81 |

2 THE MATRIX ASSOCIATED WITH A LINEAR MAP | 82 |

3 BASES MATRICES AND LINEAR MAPS | 87 |

Scalar Products and Orthogonality | 95 |

2 ORTHOGONAL BASES POSITIVE DEFINITE CASE | 103 |

3 APPLICATION TO LINEAR EQUATIONS THE RANK | 113 |

4 BILINEAR MAPS AND MATRICES | 118 |

5 GENERAL ORTHOGONAL BASES | 123 |

6 THE DUAL SPACE AND SCALAR PRODUCTS | 125 |

7 QUADRATIC FORMS | 132 |

8 SYLVESTERS THEOREM | 135 |

Determinants | 140 |

2 EXISTENCE OF DETERMINANTS | 143 |

3 ADDITIONAL PROPERTIES OF DETERMINANTS | 150 |

4 CRAMERS RULE | 157 |

5 TRIANGULATION OF A MATRIX BY COLUMN OPERATIONS | 161 |

6 PERMUTATIONS | 163 |

7 EXPANSION FORMULA AND UNIQUENESS OF DETERMINANTS | 168 |

4 DIAGONALIZATION OF A SYMMETRIC LINEAR MAP | 218 |

5 THE HERMITIAN CASE | 225 |

6 UNITARY OPERATORS | 227 |

Polynomials and Matrices | 231 |

2 POLYNOMIALS OF MATRICES AND LINEAR MAPS | 233 |

Triangulation of Matrices and Linear Maps | 237 |

2 THEOREM OF HAMILTONCAYLEY | 241 |

3 DIAGONALIZATION OF UNITARY MAPS | 243 |

Polynomials and Primary Decomposition | 245 |

2 GREATEST COMMON DIVISOR | 248 |

3 UNIQUE FACTORIZATION | 251 |

4 APPLICATION TO THE DECOMPOSITION OF A VECTOR SPACE | 255 |

5 SCHURS LEMMA | 260 |

6 THE JORDAN NORMAL FORM | 262 |

Convex Sets | 268 |

2 SEPARATING HYPERPLANES | 270 |

3 EXTREME POINTS AND SUPPORTING HYPERPLANES | 272 |

4 THE KREINMILMAN THEOREM | 274 |

Complex Numbers | 277 |

Iwasawa Decomposition and Others | 283 |

293 | |

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### Common terms and phrases

Assume bilinear called Chapter characteristic polynomial coefficients column vectors complex numbers components compute consisting of eigenvectors convex set coordinate vector Corollary define definite scalar product denote diagonal elements diagonal matrix dimension dimensional vector space direct sum dot product eigenspaces eigenvalues eigenvector equal Example EXERCISES exist numbers extreme point field finite dimensional vector given greatest common divisor Hence hyperplane induction invertible isomorphism j-th kernel Lemma Let F Let g linear combination linear equations linear map linearly independent linearly independent elements m x n matrix associated n x n matrix n-tuples non-zero obtain orthogonal basis orthonormal basis permutation perpendicular positive definite scalar properties quadratic form real numbers real unitary root satisfying Show space of solutions square matrix subset subspace Suppose surjective symmetric linear map symmetric matrix Theorem 2.1 triangle uniquely determined unit vectors unitary matrix verify write xnvn zero