Course Description: Applied Linear Algebra for AI and Machine Learning

Course Overview

This one-semester college course provides a comprehensive introduction to linear algebra with a strong emphasis on practical applications in artificial intelligence, machine learning, and computer vision. Students will develop both theoretical understanding and hands-on skills through interactive microsimulations that bring abstract mathematical concepts to life.

Linear algebra forms the mathematical backbone of modern AI systems. From the matrix operations that power neural networks to the transformations that enable computer vision, understanding linear algebra is essential for anyone working in data science, machine learning, or AI engineering. This course bridges the gap between abstract mathematics and real-world applications, showing students exactly how vectors, matrices, and linear transformations drive the technologies shaping our world.

Prerequisites

College Algebra or equivalent
Basic programming experience (Python recommended)
Familiarity with calculus concepts (derivatives and integrals)

Learning Objectives

By the end of this course, students will be able to:

Perform fundamental vector and matrix operations with confidence
Understand and apply linear transformations in multiple contexts
Decompose matrices using eigenvalue, SVD, and other factorization techniques
Apply linear algebra concepts to solve machine learning problems
Understand how neural networks use matrix operations for learning
Implement linear algebra algorithms for image processing and computer vision
Analyze real-world applications in autonomous systems and genertic AI

Course Structure

The course is divided into four major parts spanning 15 weeks, with each chapter containing interactive microsimulations to reinforce concepts.

Part 1: Foundations of Linear Algebra (Weeks 1-4)

Chapter 1: Vectors and Vector Spaces

An introduction to vectors as the fundamental building blocks of linear algebra. Students explore vector operations, geometric interpretations, and the concept of vector spaces.

Topics: - Vectors in 2D and 3D space - Vector addition and scalar multiplication - Dot products and cross products - Vector norms and distances - Linear combinations and span - Linear independence - Basis vectors and coordinate systems

Applications: Feature vectors in machine learning, word embeddings, representing data points in high-dimensional spaces.

Chapter 2: Matrices and Matrix Operations

Building on vectors, this chapter introduces matrices as collections of vectors and explores the rich algebra of matrix operations.

Topics: - Matrix notation and terminology - Matrix addition and scalar multiplication - Matrix-vector multiplication - Matrix-matrix multiplication - Transpose and symmetric matrices - Special matrices (identity, diagonal, triangular, orthogonal) - Matrix inverses

Applications: Data representation, adjacency matrices in graphs, transformation matrices in computer graphics.

Chapter 3: Systems of Linear Equations

Students learn to formulate and solve systems of linear equations, a fundamental skill with applications across all quantitative fields.

Topics: - Representing systems as matrix equations - Gaussian elimination - Row echelon form and reduced row echelon form - Existence and uniqueness of solutions - Homogeneous systems - Computational considerations and numerical stability

Applications: Solving optimization problems, balancing chemical equations, network flow analysis.

Chapter 4: Linear Transformations

This chapter reveals how matrices represent transformations, connecting algebraic operations to geometric intuition.

Topics: - Functions between vector spaces - Matrix representation of transformations - Rotation, scaling, shearing, and projection - Composition of transformations - Kernel and range of a transformation - Invertible transformations - Change of basis

Applications: Computer graphics transformations, coordinate system changes, feature transformations in ML pipelines.

Part 2: Advanced Matrix Theory (Weeks 5-8)

Chapter 5: Determinants and Matrix Properties

Determinants reveal fundamental properties of matrices and transformations, with applications in solving systems and computing volumes.

Topics: - Definition and computation of determinants - Properties of determinants - Geometric interpretation (area and volume scaling) - Cramer's rule - Determinants and invertibility - Computational methods for large matrices

Applications: Computing volumes in higher dimensions, checking matrix invertibility, understanding transformation behavior.

Chapter 6: Eigenvalues and Eigenvectors

One of the most important topics in linear algebra, eigenanalysis reveals the intrinsic structure of linear transformations.

Topics: - Definition of eigenvalues and eigenvectors - Characteristic polynomial - Finding eigenvalues and eigenvectors - Diagonalization - Complex eigenvalues - Spectral theorem for symmetric matrices - Power iteration method

Applications: Principal Component Analysis (PCA), Google's PageRank algorithm, stability analysis of dynamical systems.

Chapter 7: Matrix Decompositions

Matrix factorizations provide powerful tools for analysis, computation, and dimensionality reduction.

Topics: - LU decomposition - QR decomposition - Cholesky decomposition - Singular Value Decomposition (SVD) - Low-rank approximations - Numerical considerations

Applications: Recommender systems, image compression, solving least squares problems, noise reduction.

Chapter 8: Vector Spaces and Inner Product Spaces

Generalizing concepts from earlier chapters, this section develops the abstract framework underlying all applications.

Topics: - Abstract vector spaces - Subspaces and their properties - Inner products and norms - Orthogonality and orthonormal bases - Gram-Schmidt orthogonalization - Projections and least squares - Fundamental subspaces of a matrix

Applications: Signal processing, function approximation, optimization in machine learning.

Part 3: Linear Algebra in Machine Learning (Weeks 9-12)

Chapter 9: Linear Algebra Foundations of Machine Learning

This chapter explicitly connects linear algebra concepts to core machine learning algorithms and techniques.

Topics: - Data as matrices: features and observations - Covariance matrices and correlation - Principal Component Analysis (PCA) in depth - Linear regression as matrix equations - Regularization: Ridge and Lasso - Gradient descent in matrix form - Batch processing with matrix operations

Applications: Feature extraction, dimensionality reduction, predictive modeling, data preprocessing.

Chapter 10: Neural Networks and Deep Learning

Neural networks are fundamentally sequences of linear transformations interleaved with nonlinearities. This chapter reveals the matrix mathematics powering deep learning.

Topics: - Neurons as linear functions with activation - Weight matrices and bias vectors - Forward propagation as matrix multiplication - Backpropagation and the chain rule with matrices - Convolutional layers as structured matrix operations - Batch normalization and layer normalization - Attention mechanisms and transformer architecture - Tensor operations and higher-order arrays

Applications: Image classification, natural language processing, speech recognition, recommendation systems.

Chapter 11: Generative AI and Large Language Models

Modern generative AI systems rely heavily on sophisticated linear algebra. This chapter explores the mathematical foundations of these transformative technologies.

Topics: - Embedding spaces and semantic similarity - Attention mechanisms as matrix operations - Key, Query, and Value matrices in transformers - Self-attention and cross-attention - Position encodings - Linear projections in multi-head attention - Low-rank adaptations (LoRA) for fine-tuning - Matrix factorization in generative models - Latent spaces and interpolation

Applications: Large language models (GPT, Claude), image generation (Stable Diffusion, DALL-E), text-to-speech systems.

Chapter 12: Optimization and Learning Algorithms

Optimization is the engine of machine learning, and linear algebra provides the tools to understand and improve optimization algorithms.

Topics: - Gradient vectors and Hessian matrices - Convexity and positive definite matrices - Newton's method and quasi-Newton methods - Stochastic gradient descent - Momentum and adaptive learning rates (Adam, RMSprop) - Second-order optimization methods - Constrained optimization with Lagrange multipliers - Conditioning and numerical stability

Applications: Training neural networks, hyperparameter optimization, constrained learning problems.

Part 4: Computer Vision and Autonomous Systems (Weeks 13-15)

Chapter 13: Image Processing and Computer Vision

Images are matrices of pixel values, making linear algebra the natural language for image processing and computer vision.

Topics: - Images as matrices and tensors - Convolution as matrix operations - Image filtering (blur, sharpen, edge detection) - Fourier transforms and frequency domain - Image compression using SVD - Color spaces and transformations - Feature detection and description - Homography and perspective transformations

Applications: Photo editing, medical imaging, satellite imagery analysis, facial recognition.

Chapter 14: 3D Geometry and Transformations

Understanding 3D geometry is essential for robotics, augmented reality, and autonomous vehicles. This chapter covers the linear algebra of 3D transformations.

Topics: - 3D coordinate systems - Rotation matrices and Euler angles - Quaternions and rotation representation - Homogeneous coordinates - Rigid body transformations - Camera models and projection matrices - Stereo vision and triangulation - Point cloud processing

Applications: Robotics, augmented reality, 3D reconstruction, motion capture.

Chapter 15: Autonomous Driving and Sensor Fusion

The capstone chapter applies all course concepts to the complex, safety-critical domain of autonomous vehicles.

Topics: - LIDAR point cloud processing - Camera calibration and rectification - Sensor fusion with Kalman filters - State estimation and prediction - Simultaneous Localization and Mapping (SLAM) - Object detection and tracking - Path planning with linear constraints - Safety-critical computation considerations

Applications: Self-driving cars, drone navigation, warehouse robots, autonomous delivery systems.

Interactive Microsimulations

Each chapter includes interactive microsimulations that allow students to:

Visualize abstract concepts in 2D and 3D
Experiment with parameters and see immediate results
Build intuition through hands-on exploration
Connect mathematical formulas to visual representations
Practice computational skills in a forgiving environment

Example microsimulations include:

Vector Operations Playground: Add, subtract, and scale vectors interactively
Matrix Transformation Visualizer: See how matrices transform shapes in 2D
Eigenvalue Explorer: Watch eigenvectors remain on their span during transformation
SVD Image Compressor: Adjust rank and see image quality vs. compression tradeoffs
Neural Network Forward Pass: Step through matrix multiplications in a simple network
Attention Mechanism Visualizer: See how attention weights are computed
Kalman Filter Tracker: Fuse noisy sensor measurements in real-time
PCA Dimension Reducer: Project high-dimensional data and see variance preserved

Assessment

Weekly Problem Sets (30%): Analytical and computational problems
Microsimulation Labs (20%): Hands-on exploration with written reflections
Midterm Exam (20%): Covering Parts 1 and 2
Final Project (30%): Apply linear algebra to a real-world problem in ML, computer vision, or autonomous systems

Required Materials

Textbook: This interactive intelligent textbook with embedded microsimulations
Software: Python with NumPy, Matplotlib, and scikit-learn
Optional: GPU access for deep learning exercises

Target Audience

This course is designed for:

Computer Science majors pursuing AI/ML specializations
Data Science students seeking mathematical foundations
Engineering students interested in robotics and autonomous systems
Applied Mathematics students wanting practical applications
Graduate students needing linear algebra foundations for research

Why This Course Matters

Linear algebra is not just a prerequisite checkbox—it is the language in which modern AI systems are written. Understanding matrices and transformations at a deep level enables you to:

Debug ML models by understanding what's happening mathematically
Optimize performance by choosing efficient matrix operations
Innovate by seeing new ways to apply linear algebra concepts
Communicate with researchers and engineers using shared mathematical vocabulary
Adapt to new techniques that build on these foundations

The future of technology is built on linear algebra. This course gives you the tools to be part of building that future.

Learning Objectives Sorted by Bloom's Taxonomy

The following learning objectives are organized according to the 2001 revised Bloom's Taxonomy, progressing from foundational cognitive skills to higher-order thinking. Each level builds upon the previous, ensuring students develop comprehensive mastery of applied linear algebra.

Remember

At this foundational level, students will retrieve and recall essential facts, terminology, and procedures.

Define key terms including vector, matrix, scalar, transpose, determinant, eigenvalue, and eigenvector
List the properties of matrix operations (associativity, distributivity, non-commutativity of multiplication)
Identify special matrix types: identity, diagonal, symmetric, orthogonal, positive definite, and sparse
Recall the conditions for matrix invertibility
State the definition of linear independence and span
Recognize the notation for vector norms (L1, L2, L-infinity)
Name the four fundamental subspaces of a matrix
List the steps of Gaussian elimination
Identify the components of SVD: U, Σ, and V matrices
Recall the structure of neural network layers (weights, biases, activations)
State the formula for computing attention scores in transformers
Recognize common matrix decomposition types (LU, QR, Cholesky, SVD)

Understand

At this level, students will demonstrate comprehension by explaining concepts and interpreting mathematical relationships.

Explain the geometric interpretation of the dot product as projection
Describe how matrix multiplication represents composition of linear transformations
Interpret the meaning of eigenvalues as scaling factors along eigenvector directions
Summarize how SVD decomposes a matrix into rotations and scaling
Explain why the determinant represents the volume scaling factor of a transformation
Describe the relationship between the rank of a matrix and its solution space
Interpret covariance matrices in terms of data spread and correlation
Explain how PCA uses eigenvectors to find principal components
Describe how gradient descent uses the gradient vector to minimize loss functions
Summarize the role of weight matrices in neural network forward propagation
Explain how attention mechanisms compute relevance between tokens using dot products
Describe the purpose of the Kalman filter in combining predictions with measurements
Interpret homogeneous coordinates and their role in projective geometry

Apply

Students will use learned procedures and concepts to solve problems in familiar and new contexts.

Perform matrix-vector and matrix-matrix multiplication by hand and programmatically
Solve systems of linear equations using Gaussian elimination and matrix inverses
Compute eigenvalues and eigenvectors for 2×2 and 3×3 matrices
Apply the Gram-Schmidt process to orthogonalize a set of vectors
Calculate the SVD of a matrix and use it for low-rank approximation
Implement PCA to reduce dimensionality of a dataset
Use matrix calculus to compute gradients for optimization problems
Apply linear regression using the normal equations
Implement forward propagation through a neural network layer
Construct rotation and transformation matrices for 2D and 3D graphics
Apply convolution kernels to perform image filtering operations
Use homography matrices to correct perspective in images
Implement the power iteration method to find dominant eigenvalues

Analyze

Students will break down complex systems into components and examine relationships between parts.

Analyze the conditioning of a matrix and its impact on numerical stability
Decompose the behavior of a linear transformation into its action on eigenspaces
Examine the tradeoffs between different matrix decomposition methods for specific applications
Analyze how the choice of basis affects the representation of linear transformations
Compare the computational complexity of direct vs. iterative methods for solving linear systems
Investigate how regularization terms modify the solution space in linear regression
Analyze the information flow through neural network layers using matrix dimensions
Examine how attention patterns reveal relationships in transformer models
Analyze the effect of different kernel sizes and strides on convolutional layer outputs
Decompose a camera projection matrix into intrinsic and extrinsic parameters
Analyze sensor fusion algorithms to understand how different data sources are weighted
Examine the stability of dynamical systems through eigenvalue analysis
Investigate the relationship between matrix rank and the information preserved in compression

Evaluate

Students will make judgments and decisions based on criteria, standards, and evidence.

Assess the numerical stability of different algorithms for computing matrix inverses
Evaluate the appropriate rank for SVD truncation based on reconstruction error and compression ratio
Judge the suitability of different dimensionality reduction techniques for specific datasets
Critique the choice of optimization algorithms based on problem characteristics (convexity, scale, sparsity)
Evaluate the effectiveness of different regularization strategies for preventing overfitting
Assess the tradeoffs between model complexity and interpretability in linear models
Judge the quality of learned embeddings based on semantic similarity measures
Evaluate different attention mechanisms for computational efficiency and performance
Assess the accuracy of camera calibration by analyzing reprojection errors
Critique sensor fusion approaches based on noise characteristics and update rates
Evaluate path planning solutions based on optimality and computational constraints
Judge the robustness of SLAM algorithms under different environmental conditions
Assess when to use dense vs. sparse matrix representations based on memory and speed requirements

Create

At the highest cognitive level, students will synthesize knowledge to design, construct, and develop novel solutions.

Design a complete data preprocessing pipeline using linear algebra operations
Develop a custom dimensionality reduction approach for a specific application domain
Construct a neural network architecture with appropriate layer dimensions for a given task
Create novel image filters by designing custom convolution kernels
Design a feature extraction system using learned linear projections
Develop a recommendation system using matrix factorization techniques
Construct a real-time object tracking system using Kalman filtering
Design a camera calibration procedure for a multi-camera autonomous vehicle system
Create a 3D reconstruction pipeline from stereo image pairs
Develop a sensor fusion algorithm that combines LIDAR, camera, and IMU data
Design an efficient batched matrix computation strategy for GPU acceleration
Construct an interpretable linear model that balances accuracy with explainability
Create an interactive visualization tool demonstrating linear algebra concepts
Design a complete autonomous navigation system integrating perception, localization, and planning