Linear Algebra and Geometry
Peter Koroteev, UC Berkeley
Summer Course Format:
Asyncronous lectures and prerecorded video material. Syncronous discussions and office hours.
Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product spaces. Eigenvalues and eigenvectors; linear transformations, symmetric matrices. Linear secondorder differential equations; higherorder homogeneous differential equations; linear systems of ordinary differential equations; Fourier series and partial differential equations.
Textbook: LayLayMcDonald, Linear Algebra and its Applications (5th edition) and NagleSaffSnider, Fundamentals of Differential Equations and Boundary Value Problems (9th edition). A specially priced UC Berkeley paperback edition is availablecontaining the chapters of both books needed for the course.
Outline of the Course:
Week 1
:Systems of linear equations, matrices, vectors.
[Homework 1]

Chapter 1

Week 2: Linear Maps, Matrices, Determinants.
[Homework 2]

Sections 1.71.8 and 2.1, 2.2 and Chapter 3

Week 3 : Vector spaces and subspaces, including examples of function spaces, nullspace (kernel) and column space (image) of a matrix (linear transformation), bases, coordinate systems, dimension and rank, change of basis.
[Homework 3]

Chapter 4

Week 4 : Eigenvalues and eigenvectors, the characteristic equation, diagonalization, eigenvectors and linear transformations, complex eigenvalues.
[Homework 4] 
Sections 4.44.7, 5.1  5.5

Week 5 : The Euclidean inner product on R^n, orthogonal sets, orthogonal projection, GramSchmidt process, least squares problems, applications to linear models, inner product spaces. Midterm 7/22
[Homework 5]

Sections 6.16.7, 7.1, 7.4

Week 6 : Linear secondorder ODE: homogeneous equations, inhomogeneous equations using the method of undetermined coefficients.
[Homework 6]

Nagle et. al. Sections 4.24.5

Week 7 :Systems of first order linear ODE: reduction of higher order equations to single order systems, homogeneous constant coefficient equations using eigenvalues.
[Homework 7]

Nagle et. al. Sections 9.1, 9.4–9.6

Week 8 :Fourier series. Review. Final on 8/13
[Homework 8]

Nagle et. al. Sections 10.3–10.4

Other Useful Materials:
 Videos on [linear algebra] and [differential equations] by 3Blue1Brown
 Exams archives on [Tau Beta Pi]
More Advanced Stuff:
 A nice note explaining the equivalence among various definitions of the determinant of a sqaure matrix.
 "Every vector space has a basis" is equivalent to the axiom of choice. See here for a proof.
 A note on generalized eigenspaces.
 A note on the Brachistochrone problem.
 Conway's beautiful book on quadratic forms.