# Linear Algebra and Differential Equations

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*Alexander Givental*

A co-publication of the AMS and Center for Pure and Applied Mathematics at University of California, Berkeley

This is based on the course, “Linear Algebra and Differential
Equations”, taught by the author to sophomore students at UC Berkeley.

From the Introduction: “We accept the currently acting syllabus as an
outer constraint … but otherwise we stay rather far from conventional routes.

“In particular, at least half of the time is spent to present the entire
agenda of linear algebra and its applications in the \(2D\) environment;
Gaussian elimination occupies a visible but supporting position; abstract
vector spaces intervene only in the review section. Our eye is constantly kept
on why?, and very few facts (the fundamental theorem of algebra,
the uniqueness and existence theorem for solutions of ordinary differential
equations, the Fourier convergence theorem, and the higher-dimensional Jordan
normal form theorem) are stated and discussed without proof.”

Specific material in the book is organized as follows: Chapter 1 discusses
geometry on the plane, including vectors, analytic geometry, linear
transformations and matrices, complex numbers, and eigenvalues. Chapter 2
presents differential equations (both ODEs and PDEs), Fourier series, and the
Fourier method. Chapter 3 discusses classical problems of linear algebra,
matrices and determinants, vectors and linear systems, Gaussian elimination,
quadratic forms, eigenvectors, and vector spaces. The book concludes with a
sample final exam.

This series is jointly published between the AMS and the Center for Pure and Applied Mathematics at the University of California at Berkeley (UCB CPAM).

This series is jointly published between the AMS and the Center for Pure and Applied Mathematics at the University of California at Berkeley (UCB CPAM).

#### Readership

Advanced high school students, undergraduates, and their instructors.

#### Reviews & Endorsements

The material is presented in an original, concise and economic style … [The] approach appeals immediately to the geometric intuition of the reader and seems to be fruitful for educational purposes … important facts are rigorously proved … the material devoted to determinants is presented in a beautiful and effective manner … an excellent … introduction to linear algebra with interesting examples and applications to ODEs and PDEs … The book is an original and useful introduction to the subject and shall be of use to both students and lecturers in the field.

-- Zentralblatt MATH