Introduction

Linear algebra is a core foundation for many concepts in data science, machine learning, and artificial intelligence. Understanding the mathematical principles behind algorithms is essential for optimizing and interpreting data models. This course aims to provide participants with a hands-on, applied understanding of linear algebra concepts that are commonly used in data science. The course covers essential topics such as matrix operations, eigenvalues, singular value decomposition, and vector spaces, all within the context of solving real-world data problems.

Objectives

By the end of this course, participants will:

• Understand key linear algebra concepts and their relevance to data science.
• Master matrix and vector operations and how they are applied in data manipulation and analysis.
• Gain familiarity with eigenvalues and eigenvectors, and their role in dimensionality reduction and machine learning models.
• Learn how to apply linear algebra techniques to machine learning algorithms like Principal Component Analysis (PCA), Singular Value Decomposition (SVD), and linear regression.
• Gain practical experience in using Python libraries such as NumPy and SciPy for performing linear algebra operations on real datasets.

Who Should Attend?

This course is ideal for:

• Data scientists, analysts, and engineers who wish to deepen their understanding of the mathematical foundations of data science.
• Professionals with a background in statistics or computer science who need to apply linear algebra techniques in their work.
• Students and practitioners in machine learning who wish to gain a deeper insight into the linear algebra methods used in data science models.
• Anyone looking to build stronger problem-solving skills by understanding the math behind machine learning algorithms.

Day 1: Introduction to Linear Algebra in Data Science

Morning Session: Fundamentals of Linear Algebra

• What is Linear Algebra? Overview of its significance in data science and machine learning.
• Basic concepts: Scalars, vectors, matrices, and tensors
• Vector and matrix notation: Understanding mathematical notation and operations
• Operations on vectors and matrices: Addition, scalar multiplication, dot product, and matrix multiplication
• Properties of matrices: Commutativity, associativity, distributivity, and identity matrix
• Hands-on: Basic operations using Python’s NumPy library

Afternoon Session: Matrix Transformations and Properties

• Geometrical interpretation of matrix transformations
• Types of matrices: Square matrices, diagonal matrices, identity matrices, and orthogonal matrices
• Determinants and trace of a matrix: Importance in data science and machine learning
• Inverse of a matrix: Conditions for invertibility and calculating the inverse
• Eigenvalues and eigenvectors: Introduction to the concepts and applications
• Hands-on: Computing matrix inverses and eigenvalues using NumPy

Day 2: Matrix Decomposition and Applications

Morning Session: Singular Value Decomposition (SVD)

• What is Singular Value Decomposition (SVD)? Understanding the decomposition of matrices into eigenvalues and eigenvectors
• Applications of SVD in data science: Dimensionality reduction, noise reduction, and image compression
• SVD in recommendation systems and collaborative filtering
• Hands-on: Performing SVD on datasets using NumPy and SciPy

Afternoon Session: Principal Component Analysis (PCA)

• Introduction to PCA: Concept of dimensionality reduction and the use of eigenvectors for finding principal components
• Understanding variance and how PCA retains maximum information while reducing dimensions
• Applications of PCA in machine learning and data visualization
• Principal component analysis for feature extraction in datasets
• Hands-on: Implementing PCA on real-world datasets (e.g., Iris dataset) using Scikit-learn

Day 3: Linear Systems and Regression

Morning Session: Solving Linear Systems

• Introduction to systems of linear equations and their representation as matrix equations
• Gaussian elimination and matrix row reduction techniques
• The role of linear systems in data science: Solving systems in real-world scenarios
• Understanding the rank of a matrix and its role in determining the solvability of a system
• Hands-on: Solving linear systems using Python libraries like NumPy and SciPy

Afternoon Session: Linear Regression and Optimization

• Introduction to linear regression: Modeling relationships between dependent and independent variables
• Least squares method: Minimizing the error and optimizing coefficients in a linear regression model
• Connection between linear algebra and linear regression: Solving for coefficients using matrix operations
• Gradient descent: Optimization technique for minimizing the loss function
• Hands-on: Implementing linear regression from scratch using matrix operations in Python

Day 4: Advanced Topics in Linear Algebra for Data Science

Morning Session: Eigenvalue Decomposition and Its Applications

• Detailed understanding of eigenvalue decomposition: The connection to matrix diagonalization
• Eigenvalue decomposition in machine learning: Applications in unsupervised learning, clustering, and spectral analysis
• Spectral clustering: Using eigenvectors for clustering high-dimensional data
• Hands-on: Applying eigenvalue decomposition for clustering and classification tasks

Afternoon Session: Advanced Matrix Operations and Their Applications

• Matrix factorization techniques: LU decomposition and QR decomposition
• Understanding the role of matrix factorization in solving systems of linear equations and optimization
• Applications in data science: Recommender systems, anomaly detection, and optimization problems
• Introduction to Tensor Decomposition and its use in large-scale machine learning problems
• Hands-on: Implementing QR decomposition and matrix factorization for recommendation systems

Day 5: Real-World Applications and Final Project

Morning Session: Linear Algebra in Machine Learning Algorithms

• Review of machine learning algorithms that rely heavily on linear algebra: Support Vector Machines (SVM), Logistic Regression, and Neural Networks
• Optimization techniques in machine learning: Using matrix operations for efficient training and model fitting
• The role of linear algebra in deep learning: Neural network architectures and backpropagation
• Hands-on: Building a simple machine learning model using linear algebra concepts (e.g., Logistic Regression or SVM)

Afternoon Session: Final Project and Conclusion

• Final project: Participants apply linear algebra techniques to solve a data science problem (e.g., dimensionality reduction, linear regression, PCA for feature selection)
• Group work: Implementing and evaluating a solution using Python and linear algebra methods
• Course wrap-up: Review of key concepts and techniques
• Final Q&A and discussion of next steps for learning more advanced topics in machine learning and linear algebra

Materials and Tools:

• Software and tools: Python, NumPy, SciPy, Matplotlib, Scikit-learn
• Real-world datasets for hands-on exercises (e.g., Iris, Wine dataset, stock price data)
• Recommended readings: Key chapters from “Introduction to Linear Algebra” by Gilbert Strang and related resources

Conclusion and Final Assessment

• Recap of key concepts: Matrix operations, eigenvalues/eigenvectors, PCA, SVD, and their applications in data science
• Final assessment: Project presentation and evaluation of practical knowledge
• Certification of completion for those who successfully complete the course and final project