Linear Algebra Problems Book

Description

Front Cover Title Page Contents Preface CHAPTER 1 Linear Spaces 1.0. Terminology and General Notes 1.1. Definition of Linear Space 1.2. Linear Dependence 1.3. Spans. Rank of Vector Sets 1.4. Basis and Dimension of Space 1.5. Sum and Intersectionof Subspaces CHAPTER 2 Euclidean and Unitary Spaces 2.0. Terminology .and General Notes 2.1. Definition of Euclidean Space 2.2. Orthogonality, Orthonormal Basis, Orthogonalization Procedure 2.3. Orthogonal Complement, Orthogonal Sums of Subspaces 2.4. Lengths, Angles, Distances 2.5. Unitary Spaces CHAPTER 3 Determinants 3.0. Terminology aod General Notes 3.1. Evaluation and the Simplest Properties of Determinants 3.2. Minors, Cofactors and the Laplace Theorem 3.3. Determinants and the Volume of a Parallelepiped in a Euclidean Space 3.4. Computing the Determinants by the Elimination Method CHAPTER 4 Systems of Linear Equations 4.0. Terminology and General Notes 4.1. The Rank of a Matrix 4.2. Planes in a Linear Space 4.3. Planes in a Euclidean Space 4.4. Homogeneous Systems of Linear Equations 4.5. Nonhomogeneous Systems of Linear Equations CHAPTER 5 Linear Operators and Matrices 5.0. Terminology and General Notes 5.1. The Definition of a Linear Operator, the Image and Kernel of an Operator 5.2. Linear Operations over Operators 5.3. Multiplication of Operators 5.4. Operations over Matrices 5.5. The Inverse of a Matrix 5.6. The Matrix of a Linear Operator, Transfer to Another Basis, Equivalent and Similar Matrices CHAPTER 6 Linear Operator Structure 6.0. Terminology and General Notes 6.1. Eigenvalues and Eigenvectors 6.2. The Characterlstic Polynomial 6.3. Invariant Subspaees 6.4. Root Subspaces and the Jordan Form CHAPTER 7 Unitary Space Operators 7.0. Terminology and General Notes 7.1. Conjugate Operator. Conjugate Matrix 7.2. Normal Operators and Matrices 7.3. Unitary Operators and Matrices 7.4. Hermitian Operators and Matrices 7.5. Positive-Semi definite and Positive-Definite Operators and Matrices 7.6. Singular Values and the Polar Representation 7.7. Hermitian Decomposition 7.8. Pseudosolutions and Pseudoinverse Operators 7.9. Quadratic Forms CHAPTER 8 Metric Problems in Linear Space 8.0. Terminology and General Notes 8.1. Normed Linear Space 8.2. Norms of Operators and Matrices 8.3. Matrix Norms and Systems of Linear Equations 8.4. Matrix Norms and Eigenvalues Hints Answers and Solutions Index