Linear Algebra
Towards a Coordinate-Free Approach
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Beschreibung
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.
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.
Presents linear algebra through a fully coordinate-free and geometric framework Develops multilinear algebra, exterior algebra, and the Hodge star operator in depth Bridges elementary linear algebra and advanced studies in geometry and theoretical physics
Autor*in
Paul Dayao
Themen in »Linear Algebra«
Multi-Linear Algebra Spectral Theory Operator Theory Advanced Linear Algebra