Tensors are ubiquitous in the sciences. One reason for their ubiquity is
that they provide a useful way to organize data. Geometry is a powerful
tool for extracting information from data sets, and a beautiful subject in its
own right. This book has three intended uses: as a classroom textbook, a
reference work for researchers, and a research manuscript.
0.1. Usage
Classroom uses. Here are several possible courses one could give from this
(1) The first part of this text is suitable for an advanced course in
multilinear algebra—it provides a solid foundation for the study of
tensors and contains numerous applications, exercises, and exam-
ples. Such a course would cover Chapters 1–3 and parts of Chapters
(2) For a graduate course on the geometry of tensors not assuming
algebraic geometry, one can cover Chapters 1, 2, and 4–8 skipping
§§2.9–12, 4.6, 5.7, 6.7 (except Pieri), 7.6 and 8.6–8.
(3) For a graduate course on the geometry of tensors assuming alge-
braic geometry and with more emphasis on theory, one can follow
the above outline only skimming Chapters 2 and 4 (but perhaps
add §2.12) and add selected later topics.
(4) I have also given a one-semester class on the complexity of ma-
trix multiplication using selected material from earlier chapters and
then focusing on Chapter 11.
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