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This is a translation of the text based on a lecture series delivered by Professor S.S Chern in Beijing University in 1980. The contribution to mathematics literature combines simplicity and economy of approach with depth of contents.

From Tzuong-Tsieng Moh, a seasoned expert in algebra, comes a new book for students to better understand linear algebra. Writing from an experienced standpoint, Moh covers the many standard aspects comprising linear algebra, such as echelon forms, matrix algebra, linear transformations, and more. Moh further includes several advanced topics and applications, as well as self-correcting codes, Heisenberg's uncertainty principle, Maxwell's equations in relativity form, Google's search engine, and the theory of finitely generated modules over a PID. This book is ideal for both newcomers and experienced readers who want to attain a deeper understanding on both the basics and advanced topics of linear algebra and its vast applications. The wide range of topics combined with the depth of each discussion make it essential to be on the shelf of every mathematical beginner and enthusiast.

An introduction to the topics in number theory related to the central tools - which are based on the estimates of character sums and the estimates of the eigenvalues of Hecke operators - with the ultimate goal of presenting their applications. Topics covered include L-functions and modular forms.