Buch, Englisch, 320 Seiten, Format (B × H): 155 mm x 235 mm
Towards a Coordinate-Free Approach
Buch, Englisch, 320 Seiten, Format (B × H): 155 mm x 235 mm
Reihe: Springer Asia Pacific Mathematics Series
ISBN: 978-981-9270-68-2
Verlag: Springer
This book presents a coordinate-free approach to linear algebra, emphasizing the geometric and structural foundations of the subject rather than matrix-based computation. While introductory courses often focus on matrices, row reduction, and systems of linear equations, this text develops linear algebra through linear maps, operator theory, and multilinear geometry, highlighting concepts that are independent of any particular choice of coordinates.
The book motivates the need for a coordinate-free formulation by showing its importance in modern mathematics and theoretical physics, where global coordinate systems may not exist. Fundamental concepts such as determinants, traces, adjoints, and eigenvalues are developed from an intrinsic viewpoint. To maintain a coherent geometric perspective, algorithmic matrix techniques are largely omitted, allowing the focus to remain on the underlying structures of linear algebra.
The first part of the book develops the theory of vector spaces, linear maps, eigenvalues, diagonalization, inner product spaces, self-adjoint operators, Jordan canonical form, trace, determinant, and singular value decomposition. Matrices are presented as representations of linear operators rather than primary objects. The second part introduces multilinear algebra, including tensor products, exterior algebras, metric tensors, musical isomorphisms, and the Hodge star operator.
Designed for advanced undergraduate students, beginning graduate students, and researchers, the book serves as a bridge between standard linear algebra and more advanced subjects such as differential geometry, functional analysis, and theoretical physics. Carefully selected exercises help readers develop both conceptual understanding and mathematical maturity. By treating linear algebra as the study of structures and transformations rather than coordinate calculations, this book provides a rigorous foundation for modern geometry and mathematical physics.
Zielgruppe
Upper undergraduate
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
Part I - Linear Algebra: Chapter 1: Vector Spaces.- Chapter 2: Linear Maps.- Chapter 3: Matrices and Linear Maps.- Chapter 4: Polynomials and Eigenvectors.- Chapter 5: Diagonalization.- Chapter 6: Inner Product Spaces.- Chapter 7: Operators on Inner Product Spaces.- Chapter 8: Generalized Eigenvectors and the Jordan Form.- Chapter 9: Trace and Determinant.- Chapter 10: Singular Value Decomposition.- Part II- Multilinear Algebra: Chapter 11: Multilinear Maps and Exterior Geometry.
