Linear Algebra and Geometry
Peter Koroteev, UC Berkeley
Summer Course Format:
Eight
hours of lecture per week and prerecorded video material.
Description:
Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product spaces. Eigenvalues and eigenvectors; linear transformations, symmetric matrices.
Textbook:
Lay-Lay-McDonald,
Linear Algebra and its Applications (5th edition).
Outline of the Course:
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Week 1
:
Complex numbers, vector spaces, subspaces.
[Homework 1] due on 7/13
Videos of lectures:
[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: [Week 1]
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Chapter 1
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Week 2
: Linear Maps, Matrices, Determinants.
[Homework 2] due on 7/20
Videos of lectures:
[Lecture 1- Homogeneous and Inhomogeneous Systems, Spans]
[Video 2- Matrix Operations]
[Video 3- Linear Subspaces]
[Video 4- Determinants]
[Discussion]
Slides:
[Linear Subspace]
[Matrix Multiplication]
[Determinants]
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Sections 1.7-1.8 and 2.1, 2.2
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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. Midterm on 7/25.
[Homework 3] due on 7/27
Videos of lectures:
[Determinants-minor expansion]
[Vector Spaces]
[Linear Independence, Bases, Inverse Functions]
Slides:
[Determinants-minor expansion]
[Vector Spaces, Subspaces]
[Bases, Inverse Functions]
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Sections 3.3, 4.1-4.3
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Week 4
: Eigenvalues and eigenvectors, the characteristic equation, diagonalization, eigenvectors and linear transformations, complex eigenvalues.
[Homework 4] due on 8/03
Videos of lectures:
[Change of Basis]
[Eigenvectors and Eigenvalues I]
[Eigenvectors and Eigenvalues II]
[Complex Numbers, Orthogonality]
[Discussion]
Slides:
[Change of Bases]
[Eigenvectors and Eigenvalues I]
[Eigenvectors and Eigenvalues II]
[Complex Numbers, Orthogonality]
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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. Final 8/8
[Homework 5] due on 8/9
Videos of lectures:
[Orthnormal Basis]
[Orthogonal Projection]
[Least Squares, Orthogonality]
[Singular Value Decomposition]
Slides:
[Orthonormal Basis]
[Orthogonal Projection]
[Least Squares, Orthogonality]
[Singular Value Decomposition]
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Sections 6.1-6.7, 7.1, 7.4
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