Explore an extensive 19-hour linear algebra course covering a wide range of topics from fundamental concepts to advanced applications. Dive into singular value decomposition (SVD), matrix operations, linear systems, regression, principal component analysis (PCA), Fourier analysis, compressed sensing, and more. Learn through a combination of theoretical explanations and practical implementations using MATLAB and Python. Gain hands-on experience with image compression, eigenfaces, data denoising, PDE solving, and various signal processing techniques. Discover the power of sparsity in data representation and its applications in compressed sensing and robust regression. Enhance your understanding of linear algebra and its real-world applications in data science, machine learning, and engineering.
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
