QR Decomposition
Understand the QR Decomposition, its mathematical foundation, applications, and how to compute it. Learn how QR Decomposition is used in solving linear systems, least squares problems, and more.
LU Decomposition
Understand LU Decomposition, its mathematical foundation, and applications. Learn how to compute LU Decomposition and use it for solving linear systems, inverting matrices, and more.
Matrix Factorizations in Numerical Analysis
Explore various matrix factorizations, including LU, Cholesky, and QR decompositions, and their applications in numerical analysis for solving linear systems, eigenvalue problems, and more.
Norms (Matrix and Vector)
Explore matrix and vector norms, their definitions, types, and applications in linear algebra. Learn how norms are used to measure the size of vectors and matrices and their significance in data science and numerical analysis.
Positive Definite Matrices
Understand the concept of positive definite matrices, their properties, and their significance in linear algebra, optimization, and statistics. Learn how to test for positive definiteness and explore practical applications.
Condition Numbers in Linear Systems
Explore the concept of condition numbers in linear systems, their significance in numerical stability, and how they are calculated using matrix norms. Understand the impact of condition numbers on the accuracy and reliability of solutions.
Solving Over- and Under-Determined Systems
Explore methods for solving over-determined and under-determined systems in linear algebra, including Least Squares solutions, regularization techniques, and practical applications in data science and engineering.
Least Squares and Regression
Explore the Least Squares method and its application in regression analysis, including linear regression, the normal equations, and the use of matrix algebra in fitting models to data.
Introduction to Linear Programming
Explore the fundamentals of linear programming, including formulation of linear programs, the simplex method, duality, and practical applications in optimization and operations research.
Diagonalization and Jordan Form
A comprehensive exploration of matrix diagonalization, the Jordan canonical form, and their applications in simplifying linear transformations and solving systems of linear equations.
Spectral Decomposition
Explore Spectral Decomposition, its mathematical foundation, applications in linear algebra, and its significance in data science, physics, and engineering.
Advanced Concepts in Eigenvalues and Eigenvectors
Explore intermediate-level topics in eigenvalues and eigenvectors, including generalized eigenvectors, the Jordan Canonical Form, the Spectral Theorem, and applications in differential equations and dynamical systems.
Linear Discriminant Analysis (LDA)
Explore the mathematical foundations of Linear Discriminant Analysis (LDA), including its derivation, optimization, and application in dimensionality reduction and classification.
Singular Value Decomposition (SVD)
A detailed exploration of Singular Value Decomposition (SVD), its mathematical foundation, and its applications in data science, including dimensionality reduction and noise reduction.
Principal Component Analysis (PCA) using Linear Algebra
Explore Principal Component Analysis (PCA) through the lens of linear algebra, including its mathematical foundation, computation using eigenvectors and eigenvalues, and applications in dimensionality reduction and data science.
Kernel Principal Component Analysis (Kernel PCA)
Explore Kernel PCA, an extension of Principal Component Analysis that allows for non-linear dimensionality reduction using the kernel trick, with a detailed mathematical foundation and examples.
Optimization in Linear Algebra
Explore how linear algebra techniques are applied to solve optimization problems, including linear and quadratic programming, gradient descent, and regularization methods in machine learning.
Vector Spaces in Linear Models
Understanding how vector spaces relate to linear models, including the role of vector spaces in linear regression, model interpretation, and multicollinearity.
Advanced Applications in Data Science
Exploring advanced applications in data science that utilize concepts from linear algebra, including matrix factorization, recommendation systems, and spectral clustering.