This course on Linear Algebra aims to teach students the following learning outcomes and goals: understanding the concepts of sparsity, singular value decomposition (SVD), matrix approximation, dominant correlations, Frobenius norm, unitary transformations, least squares regression, principal component analysis (PCA), image compression, Fourier analysis, Fourier transform, fast Fourier transform (FFT), denoising data, solving partial differential equations (PDEs), compressed sensing, wavelets, sparse sensor placement optimization, robust regression, Laplace transform, and data-driven resolvent analysis. The course covers various tools and skills such as MATLAB and Python programming, SVD method, linear regression, image processing, FFT algorithms, wavelet analysis, compressed sensing techniques, and Laplace transforms. The teaching method includes theoretical overviews, mathematical explanations, practical examples, coding exercises, and application-based case studies. The intended audience for this course includes students, professionals, and researchers interested in data analysis, signal processing, image processing, machine learning, and computational mathematics.
Overview
Syllabus
A Compressed Overview of Sparsity.
2016 AIAA AVIATION Forum: Flow Control - Steve Brunton.
Singular Value Decomposition (SVD): Overview.
Singular Value Decomposition (SVD): Mathematical Overview.
Singular Value Decomposition (SVD): Matrix Approximation.
Singular Value Decomposition (SVD): Dominant Correlations.
The Frobenius Norm for Matrices.
SVD Method of Snapshots.
Matrix Completion and the Netflix Prize.
Unitary Transformations.
Linear Systems of Equations, Least Squares Regression, Pseudoinverse.
Least Squares Regression and the SVD.
Linear Systems of Equations.
Linear Regression.
Principal Component Analysis (PCA).
SVD and Optimal Truncation.
SVD: Image Compression [Matlab].
SVD: Image Compression [Python].
Unitary Transformations and the SVD [Matlab].
Unitary Transformations and the SVD [Python].
Linear Regression 1 [Matlab].
Linear Regression 2 [Matlab].
Linear Regression 1 [Python].
Linear Regression 2 [Python].
Linear Regression 3 [Python].
SVD and Alignment: A Cautionary Tale.
Principal Component Analysis (PCA) [Matlab].
Principal Component Analysis (PCA) 1 [Python].
Principal Component Analysis (PCA) 2 [Python].
SVD: Eigenfaces 1 [Matlab].
SVD: Eigenfaces 2 [Matlab].
SVD: Eigenfaces 3 [Matlab].
SVD: Eigenfaces 4 [Matlab].
SVD: Eigen Action Heros [Matlab].
SVD: Eigenfaces 3 [Python].
SVD: Eigenfaces 2 [Python].
SVD: Eigenfaces 1 [Python].
SVD: Optimal Truncation [Matlab].
SVD: Optimal Truncation [Python].
SVD: Importance of Alignment [Python].
SVD: Importance of Alignment [Matlab].
Randomized SVD Code [Matlab].
Randomized SVD Code [Python].
Randomized Singular Value Decomposition (SVD).
Randomized SVD: Power Iterations and Oversampling.
Fourier Analysis: Overview.
Fourier Series: Part 1.
Fourier Series: Part 2.
Inner Products in Hilbert Space.
Complex Fourier Series.
Fourier Series [Matlab].
Fourier Series [Python].
Fourier Series and Gibbs Phenomena [Matlab].
Fourier Series and Gibbs Phenomena [Python].
The Fourier Transform.
The Fourier Transform and Derivatives.
The Fourier Transform and Convolution Integrals.
Parseval's Theorem.
Solving the Heat Equation with the Fourier Transform.
The Discrete Fourier Transform (DFT).
Computing the DFT Matrix.
The Fast Fourier Transform (FFT).
The Fast Fourier Transform Algorithm.
Denoising Data with FFT [Matlab].
Denoising Data with FFT [Python].
Computing Derivatives with FFT [Matlab].
Computing Derivatives with FFT [Python].
Solving PDEs with the FFT [Matlab].
Solving PDEs with the FFT [Python].
Why images are compressible: The Vastness of Image Space.
What is Sparsity?.
Sparsity and Parsimonious Models: Everything should be made as simple as possible, but no simpler.
Compressed Sensing: Overview.
Compressed Sensing: Mathematical Formulation.
Compressed Sensing: When It Works.
Sparsity and the L1 Norm.
Solving PDEs with the FFT, Part 2 [Matlab].
Solving PDEs with the FFT, Part 2 [Python].
The Spectrogram and the Gabor Transform.
Spectrogram Examples [Matlab].
Spectrogram Examples [Python].
Uncertainty Principles and the Fourier Transform.
Wavelets and Multiresolution Analysis.
Image Compression and the FFT.
Sparse Sensor Placement Optimization for Reconstruction.
Sparse Sensor Placement Optimization for Classification.
Sparse Representation (for classification) with examples!.
Image Compression with Wavelets (Examples in Python).
Image Compression with the FFT (Examples in Matlab).
Image Compression and Wavelets (Examples in Matlab).
Image Compression and the FFT (Examples in Python).
Beating Nyquist with Compressed Sensing, part 2.
Underdetermined systems and compressed sensing [Matlab].
Underdetermined systems and compressed sensing [Python].
Beating Nyquist with Compressed Sensing.
Robust Regression with the L1 Norm.
Robust Regression with the L1 Norm [Matlab].
Robust Regression with the L1 Norm [Python].
Beating Nyquist with Compressed Sensing, in Python.
PySINDy: A Python Library for Model Discovery.
The Laplace Transform: A Generalized Fourier Transform.
Laplace Transforms and Differential Equations.
Laplace Transform Examples.
Sparsity and Compression: An Overview.
Data-Driven Resolvent Analysis.
Taught by
Steve Brunton
