Introduction to Linear Algebra (Classic Version)

<br> <br> 1. Matrices and Systems of Linear Equations. <br> <p> </p> <div style="margin-left: 0.2in;"> Introduction to Matrices and Systems of Linear Equations. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Echelon Form and Gauss-Jordan Elimination. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Consistent Systems of Linear Equations. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Applications (Optional). </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Matrix Operations. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Algebraic Properties of Matrix Operations. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Linear Independence and Nonsingular Matrices. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Data Fitting, Numerical Integration, and Numerical Differentiation (Optional). </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Matrix Inverses and Their Properties. </div> <p></p> <br> <br> 2. Vectors in 2-Space and 3-Space. <br> <p> </p> <div style="margin-left: 0.2in;"> Vectors in the Plane. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Vectors in Space. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> The Dot Product and the Cross Product. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Lines and Planes in Space. </div> <p></p> <br> <br> 3. The Vector Space Rn. <br> <p> </p> <div style="margin-left: 0.2in;"> Introduction. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Vector Space Properties of Rn. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Examples of Subspaces. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Bases for Subspaces. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Dimension. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Orthogonal Bases for Subspaces. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Linear Transformations from Rn to Rm. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Least-Squares Solutions to Inconsistent Systems, with Applications to Data Fitting. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Theory and Practice of Least Squares. </div> <p></p> <br> <br> 4. The Eigenvalue Problem. <br> <p> </p> <div style="margin-left: 0.2in;"> The Eigenvalue Problem for (2 x 2) Matrices. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Determinants and the Eigenvalue Problem. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Elementary Operations and Determinants (Optional). </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Eigenvalues and the Characteristic Polynomial. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Eigenvectors and Eigenspaces. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Complex Eigenvalues and Eigenvectors. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Similarity Transformations and Diagonalization. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Difference Equations; Markov Chains, Systems of Differential Equations (Optional). </div> <p></p> <br> <br> 5. Vector Spaces and Linear Transformations. <br> <p> </p> <div style="margin-left: 0.2in;"> Introduction. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Vector Spaces. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Subspaces. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Linear Independence, Bases, and Coordinates. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Dimension. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Inner-Product Spaces, Orthogonal Bases, and Projections (Optional). </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Linear Transformations. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Operations with Linear Transformations. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Matrix Representations for Linear Transformations. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Change of Basis and Diagonalization. </div> <p></p> <br> <br> 6. Determinants. <br> <p> </p> <div style="margin-left: 0.2in;"> Introduction. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Cofactor Expansions of Determinants. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Elementary Operations and Determinants. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Cramer's Rule. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Applications of Determinants: Inverses and Wronksians. </div> <p></p> <br> <br> 7. Eigenvalues and Applications. <br> <p> </p> <div style="margin-left: 0.2in;"> Quadratic Forms. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Systems of Differential Equations. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Transformation to Hessenberg Form. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Eigenvalues of Hessenberg Matrices. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Householder Transformations. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> The QR Factorization and Least-Squares Solutions. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Matrix Polynomials and the Cayley-Hamilton Theorem. </div> <p></p> <p> </p> <div style="margin-left: 0.2in;"> Generalized Eigenvectors and Solutions of Systems of Differential Equations. </div> <p></p> <br> <br> Appendix: An Introduction to MATLAB. <br> <br> <br> Answers to Selected Odd-Numbered Exercises. <br> <br> <br> Index. <br>