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Advanced Linear and Matrix Algebra (Hardcover, 1st ed. 2021)
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Advanced Linear and Matrix Algebra (Hardcover, 1st ed. 2021)
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This textbook emphasizes the interplay between algebra and geometry
to motivate the study of advanced linear algebra techniques.
Matrices and linear transformations are presented as two sides of
the same coin, with their connection motivating inquiry throughout
the book. Building on a first course in linear algebra, this book
offers readers a deeper understanding of abstract structures,
matrix decompositions, multilinearity, and tensors. Concepts draw
on concrete examples throughout, offering accessible pathways to
advanced techniques. Beginning with a study of vector spaces that
includes coordinates, isomorphisms, orthogonality, and projections,
the book goes on to focus on matrix decompositions. Numerous
decompositions are explored, including the Shur, spectral, singular
value, and Jordan decompositions. In each case, the author ties the
new technique back to familiar ones, to create a coherent set of
tools. Tensors and multilinearity complete the book, with a study
of the Kronecker product, multilinear transformations, and tensor
products. Throughout, "Extra Topic" sections augment the core
content with a wide range of ideas and applications, from the QR
and Cholesky decompositions, to matrix-valued linear maps and
semidefinite programming. Exercises of all levels accompany each
section. Advanced Linear and Matrix Algebra offers students of
mathematics, data analysis, and beyond the essential tools and
concepts needed for further study. The engaging color presentation
and frequent marginal notes showcase the author's visual approach.
A first course in proof-based linear algebra is assumed. An ideal
preparation can be found in the author's companion volume,
Introduction to Linear and Matrix Algebra.
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