Linear Algebra

<div class="c-un-numbered-headers-and-contents-list__container"> <ul> <li>* Sections denoted by an asterisk are optional.</li> </ul> <ol> <li><strong>Vector Spaces</strong> <ul> <li>1.1 Introduction</li> <li>1.2 Vector Spaces</li> <li>1.3 Subspaces</li> <li>1.4 Linear Combinations and Systems of Linear Equations</li> <li>1.5 Linear Dependence and Linear Independence</li> <li>1.6 Bases and Dimension</li> <li>1.7* Maximal Linearly Independent Subsets</li> </ul> <ul> <li>Index of Definitions</li> </ul></li> <li><strong>Linear Transformations and Matrices</strong> <ul> <li>2.1 Linear Transformations, Null Spaces, and Ranges</li> <li>2.2 The Matrix Representation of a Linear Transformation</li> <li>2.3 Composition of Linear Transformations and Matrix Multiplication</li> <li>2.4 Invertibility and Isomorphisms</li> <li>2.5 The Change of Coordinate Matrix</li> <li>2.6* Dual Spaces</li> <li>2.7* Homogeneous Linear Differential Equations with Constant Coefficients</li> </ul> <ul> <li>Index of Definitions</li> </ul></li> <li><strong>Elementary Matrix Operations and Systems of Linear Equations</strong> <ul> <li>3.1 Elementary Matrix Operations and Elementary Matrices</li> <li>3.2 The Rank of a Matrix and Matrix Inverses</li> <li>3.3 Systems of Linear Equations – Theoretical Aspects</li> <li>3.4 Systems of Linear Equations – Computational Aspects</li> </ul> <ul> <li>Index of Definitions</li> </ul></li> <li><strong>Determinants</strong> <ul> <li>4.1 Determinants of Order 2</li> <li>4.2 Determinants of Order n</li> <li>4.3 Properties of Determinants</li> <li>4.4 Summary|Important Facts about Determinants</li> <li>4.5* A Characterization of the Determinant</li> </ul> <ul> <li>Index of Definitions</li> </ul></li> <li><strong>Diagonalization</strong> <ul> <li>5.1 Eigenvalues and Eigenvectors</li> <li>5.2 Diagonalizability</li> <li>5.3* Matrix Limits and Markov Chains</li> <li>5.4 Invariant Subspaces and the Cayley–Hamilton Theorem</li> </ul> <ul> <li>Index of Definitions</li> </ul></li> <li><strong>Inner Product Spaces</strong> <ul> <li>6.1 Inner Products and Norms</li> <li>6.2 The Gram–Schmidt Orthogonalization Process and Orthogonal Complements</li> <li>6.3 The Adjoint of a Linear Operator</li> <li>6.4 Normal and Self-Adjoint Operators</li> <li>6.5 Unitary and Orthogonal Operators and Their Matrices</li> <li>6.6 Orthogonal Projections and the Spectral Theorem</li> <li>6.7* The Singular Value Decomposition and the Pseudoinverse</li> <li>6.8* Bilinear and Quadratic Forms</li> <li>6.9* Einstein's Special Theory of Relativity</li> <li>6.10* Conditioning and the Rayleigh Quotient</li> <li>6.11* The Geometry of Orthogonal Operators</li> </ul> <ul> <li>Index of Definitions</li> </ul></li> <li><strong>Canonical Forms</strong> <ul> <li>7.1 The Jordan Canonical Form I</li> <li>7.2 The Jordan Canonical Form II</li> <li>7.3 The Minimal Polynomial</li> <li>7.4* The Rational Canonical Form</li> </ul> <ul> <li>Index of Definitions</li> </ul></li> </ol> <h3>Appendices</h3> <ul> <li>A. Sets</li> <li>B. Functions</li> <li>C. Fields</li> <li>D. Complex Numbers</li> <li>E. Polynomials</li> </ul> <p>Answers to Selected Exercises</p> <p>Index</p> </div>