📄 Executive Summary
Linear Algebra: An Interactive Introduction (3rd edition) is designed for undergraduate students in mathematics, engineering, computer science, and the applied sciences who require a solid foundation in linear algebra. The text assumes a working familiarity with high school algebra, basic coordinate geometry, and basic trigonometry, but it develops concepts from the ground up without assuming prior exposure to abstract proofs or advanced matrix theory. Spanning both theoretical fundamentals and numerical practice, the textbook moves from concrete Euclidean geometry to computational matrix operations, abstract vector spaces, and applied matrix factorizations.
The curriculum progresses systematically from intuitive geometric beginnings to sophisticated structural and spectral concepts. It opens with vector arithmetic, dot and cross products, and the geometry of lines and planes in two- and three-dimensional Euclidean space, before extending these notions to n-dimensional space. The text then establishes methods for solving systems of linear equations using Gaussian and Gauss-Jordan elimination, augmented matrices, row-echelon forms, and rank calculations, complementing exact direct solutions with iterative schemes such as Jacobi's and Gauss-Seidel methods. Building on these foundations, the book introduces linear combinations, span, and linear independence to formalize the concepts of bases, dimension, subspaces, and the Rank-Nullity Theorem. Later chapters examine linear transformations, determinants, matrix inversions, and LU factorizations, advancing into eigenvalue problems, diagonalization, Gershgorin disks, the Power Method, Gram-Schmidt orthogonalization, least-squares approximations, QR factorization, and singular value decomposition. These topics culminate in the axiomatic treatment of abstract vector spaces and general inner product spaces through coordinate isomorphisms.
Alongside core theoretical principles, the textbook emphasizes concrete computational and modeling skills. Readers learn how linear systems model practical phenomena such as network flows, electrical circuits, chemical reaction balancing, Leontief open and closed economic systems, Markov chains with steady-state probability vectors, polynomial curve fitting, and computer graphics via homogeneous coordinates. A dedicated computational component introduces students to GNU Octave for coding vector routines, matrix factorizations, and iterative approximations. After working through the book, readers should be able to solve linear equations, analyze and represent linear transformations, perform factorizations, and implement matrix workflows in scientific software. The book deliberately places specialized topics—such as rigorous numerical precision analysis, exhaustive treatments of complex vector spaces, and advanced analytic proofs—outside its main scope, directing readers to specialized literature where appropriate.
