A linear algebra textbook that fosters mathematical thinking, problem solving abilities, and exposure to real world applications. It focuses on the aspects of linear algebra that are likely to have practical value to the student while not compromising the intrinsic mathematical form of the subject.

Chapter 1. Vectors.1.1 Vectors and Matrices in Engineering and Mathematics; n-Space.1.2 Dot Product and Orthogonality.1.3 Vector Equations of Lines and Planes.Chapter 2. Systems of Linear Equations.2.1 Introduction to Systems of Linear Equations.2.2 Solving Linear Systems by Row Reduction.2.3 Applications of Linear Systems.Chapter 3. Matrices and Matrix Algebra.3.1 Operations on Matrices.3.2 Inverses; Algebraic Properties of Matrices.3.3 Elementary Matrices; A Method for Finding A-1.3.4 Subspaces and Linear Independence.3.5 The Geometry of Linear Systems.3.6 Matrices with Special Forms.3.7 Matrix Factorizations; LU-Decomposition.3.8 Partitioned Matrices and Parallel Processing.Chapter 4. Determinants.4.1 Determinants; Cofactor Expansion.4.2 Properties of Determinants.4.3 Cramer's Role; Formula for A-1; Applications of Determinants.4.4 A First Look at Eigenvalues and Eigenvectors.Chapter 5. Matrix Models.5.1 Dynamical Systems and Markov Chains.5.2 Leontief Input-Output Models.5.3 Gauss-Seidel and Jacobi Iteration; Sparse Linear Systems.5.4 The Power Method; Application to Internet Search Engines.Chapter 6. Linear Transformations.6.1 Matrices as Transformations.6.2 Geometry of Linear Operators.6.3 Kernel and Range.6.4 Composition and Invertibility of Linear Transformations.6.5 Computer Graphics.Chapter 7. Dimension and Structure.7.1 Basis and Dimension.7.2 Properties of Bases.7.3 The Fundamental Spaces of a Matrix.7.4 The Dimension Theorem and Its Implications.7.5 The Rank Theorem and Its Implications.7.6 The Pivot Theorem and Its Implications.7.7 The Projection Theorem and Its Implications.7.8 Best Approximation and Least Squares.7.9 Orthonormal Bases and the Gram-Schmidt Process.7.10 QR-Decomposition; Householder Transformations.7.11 Coordinates with Respect to a Basis.Chapter 8. Diagonalization.8.1 Matrix Representations of Linear Transformations.8.2 Similarity and Diagonalizability.8.3 Orthogonal Diagonalizability; Functions of a Matrix.8.4 Quadratic Forms.8.5 Application of Quadratic Forms to Optimization.8.6 Singular Value Decomposition.8.7 The Pseudoinverse.8.8 Complex Eigenvalues and Eigenvectors.8.9 Hermitian, Unitary, and Normal Matrices.8.10 Systems of Differential Equations.Chapter 9. General Vector Spaces.9.1 Vector Space Axioms.9.2 Inner Product Spaces; Fourier Series.9.3 General Linear Transformations; Isomorphism.Appendix A. How to Read Theorems.Appendix B. Complex Numbers.Answers to Odd-Numbered Exercises.Photo Credits.Index.