Linear Algebra and Differential Equations

Description

The present text consists of 130 pages of lecture notes, including numerous pictures and exercises, for a one-semester course in Linear Algebra and Differential Equations. The notes are reasonably self-contained. In particular, prior knowledge of Multivariable Calculus is not required. Calculators are of little use. Intelligent, hands-on reading is expected instead. The lecture notes correspond to the course “Linear Algebra and Differential Equations” taught to sophomore students at UC Berkeley. We accept the currently acting syllabus as an outer constraint and borrow from the official textbooks two examples, 1 but otherwise we stay rather far from conventional routes. In particular, at least half of the time (Chapters 1 and 2) 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 (section 3.4); abstract vector spaces intervene only in the review section 3.7. Our eye is constantly kept on why?, and very few facts 2 are stated and discussed without proof. The notes were conceived with somewhat greater esteem for the subject, the teacher and the student than is traditionally anticipated. We hope that mathematics, when it bears some content, can be appreciated and eventually understood. We wish the reader to find some evidence in favor of this conjecture. Foreword 1. Geometry on the plane. 1.1. Vectors 1.1.1.Definitions. 1.1.2. Inner product. 1.1.3. Coordinates. 1.2. Analytical geometry. 1.2.1. Linear functions and staright lines. 1.2.2. Conic sections. 1.2.3. Quadratic forms. 1.3. Linear transformations and matrices. 1.3.1. Linearity. 1.3.2. Composition. 1.3.3. Inverses. 1.3.4. Matrix Zoo. 1.4. Complex numbers. 1.4.1. Definitions and geometrical interpretations. 1.4.2. The exponential function. 1.4.3. The Fundamental Theorem of Algebra. 1.5. Eigenvalues. 1.5.1. Linear systems. 1.5.2. Determinants. 1.5.3. Normal forms. Sample midterm exam 2. Differential equations. 2.1. ODE. 2.1.1. Existence and uniqueness of solutions. 2.1.2. Linear ODE systems. 2.2. Stability. 2.2.1. Partial derivatives. 2.2.2. Linearization. 2.2.3. Competing species. 2.3. PDE. 2.3.1. The heat equation. 2.3.2. Boundary value problems. 2.4. Fourier series. 2.4.1. Fourier coefficients. 2.4.2. Convergence. 2.4.3. Real even and odd functions. 2.5. The Fourier method. 2.5.1. The series solution. 2.5.2. Properties of solutions. Sample midterm exam 3. Linear Algebra. 3.1. Classical problems of linear algebra 3.2. Matrices and determinants. 3.2.1. Matrix algebra. 3.2.2. The determinant function. 3.2.3. Properties of determinants. 3.2.4. Cofactors. 3.3. Vectors and linear systems. 3.3.1. 3D and beyond. 3.3.2. Linear (in)dependence and bases. 3.3.3. Subspaces and dimension. 3.3.4. The rank theorem and applications. 3.4. Gaussian elimination. 3.4.1. Row reduction. 3.4.2. Applications. 3.5. Quadratic forms. 3.5.1. Inertia indices. 3.5.2. Least square fitting to data. 3.5.3. Orthonormal bases. 3.5.4. Orthogonal diagonalization. 3.5.5. Small oscillations. 3.6. Eigenvectors. 3.6.1. Diagonalization theorem. 3.6.2. Linear ODE systems. 3.6.3. Higher order linear ODEs. 3.7. Vector spaces. 3.7.1. Axioms and examples. 3.7.2. Error-correcting codes. 3.7.3. Linear operators and ODEs. 3.7.4. The heat equation revisited. Sample final exam.