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.