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:
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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
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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
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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
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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
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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
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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
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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
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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
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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.