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 second-order differential equations; higher-order homogeneous differential equations; linear systems of ordinary differential equations; Fourier series and partial differential equations.

Textbook: Lay-Lay-McDonald, Linear Algebra and its Applications (5th edition) and Nagle-Saff-Snider, 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]
Intro, Systems of Linear Equations [s] [v]
Row Reduction, Echelon Form, Vectors [s] [v]
Matrix-vector Product, Linear Independence [s] [v]
Linear Transformations [s] [v]

Videos of 3Blue1Brown: [Vectors] [Linear Combinations, Span, and Basis Vectors]
Videos of lectures (2020): [Lecture 1-Introduction, Dimensions] [Lecture 2-Systems of Linear Equations, Matrices] [Lecture 3-Row Reduction, Echelon Form] [Lecture 4-Homogeneous and Inhomogeneous Systems]
iPad lecture notes and slides (2020): [Week 1]

Chapter 1

Week 2: Linear Maps, Matrices, Determinants.

[Homework 2]
Operations on Matrices [s] [v]
Linear Subspaces [s] [v]
Determinants - Formal Properties [s] [v]
Determinants - Minor Expansion [s] [v]

Videos of 3Blue1Brown: [Matrix Multiplication as Composition] [Three-Dimensional Linear Transformations] [Determinants]

Sections 1.7-1.8 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]
Vector Spaces, Subspaces [s] [v]
Linear Independence, Bases, Inverse Functions [s] [v]

Videos of 3Blue1Brown: [Abstract Vector Spaces]

Chapter 4

Week 4 : Eigenvalues and eigenvectors, the characteristic equation, diagonalization, eigenvectors and linear transformations, complex eigenvalues.

[Homework 4]
Change of Basis [s] [v]
Eigenvectors and Eigenvalues I [v] [v]
Eigenvectors and Eigenvalues II [s] [v]
Complex Numbers, Orthogonality [s] [v]

Videos of 3Blue1Brown: [Change of Basis] [Eigenvectors and Eigenvalues] [Dot Product and Duality]

Sections 4.4-4.7, 5.1 - 5.5

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

[Homework 5]
Orthnormal Basis [s] [v]
Orthogonal Projection [s] [v]
Least Squares, Orthogonality [s] [v]
Singular Value Decomposition [s] [v]

Sections 6.1-6.7, 7.1, 7.4

Week 6 : Linear second-order ODE: homogeneous equations, inhomogeneous equations using the method of undetermined coefficients.

[Homework 6]
First and second order differential equations [s] [v]
Initial value problem, inhomogeneous problem [s] [v]
Higher order differential equations [s] [v]
Systems of ODEs [s]

Videos of 3Blue1Brown: [ODEs Intro]

Nagle et. al. Sections 4.2-4.5

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]
Systems of ODEs [s] [v]
Partial differential equations [s] [v]
Fourier Series [s] [v]
Heat Equation [s] [v]

Videos of 3Blue1Brown: [What is a PDE?] [Solving the heat equaiton]

Nagle et. al. Sections 9.1, 9.4–9.6

Week 8 :Fourier series. Review. Final on 8/13

[Homework 8]

Videos of 3Blue1Brown: [What is a Fourier Series] [Fourier Series Animation]

Nagle et. al. Sections 10.3–10.4

Other Useful Materials:

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.