Introduction to Linear Algebra Course Syllabus

This comprehensive linear algebra course provides a rigorous foundation in vector spaces, matrices, and linear transformations. Through 10 weeks of structured learning, you will master fundamental concepts essential for machine learning, computer graphics, quantum computing, and engineering applications. Blending theoretical understanding with computational practice using Python and MATLAB, this course builds unshakable mathematical reasoning alongside practical problem-solving skills for modern STEM careers.

Module 1: Vectors, Vector Spaces, and Subspaces

Establish the foundations of linear algebra by exploring vectors as geometric objects and algebraic entities. Learn vector operations, dot products, and the concept of vector spaces and subspaces. Understand linear combinations, span, and linear independence—the building blocks for all subsequent topics. This module emphasizes both geometric intuition and formal definitions.

• Vector geometry and algebraic operations (addition, scalar multiplication)
• Dot products, norms, and orthogonality concepts
• Vector spaces and subspace axioms
• Linear independence, basis, and dimension
• Span and spanning sets
• Orthogonal projections and Gram-Schmidt process

Estimated time: 25 hours

Module 2: Matrices and Matrix Algebra

Transition from abstract vector theory to practical matrix representations. Master matrix operations (multiplication, transpose, inversion), special matrix types, and determinants. Understand the fundamental theorem linking matrix rank to solutions of linear systems. Develop computational fluency in matrix manipulation and the geometric meaning behind matrix operations.

• Matrix notation and matrix operations (addition, multiplication, transpose)
• Special matrices (identity, diagonal, symmetric, orthogonal)
• Matrix rank and nullity
• Determinants: definition, properties, and computation methods
• Matrix inverses and the invertible matrix theorem
• Computational practice with MATLAB/Python

Estimated time: 28 hours

Module 3: Solving Linear Systems

Apply matrix theory to systematically solve linear systems of equations. Learn Gaussian elimination, LU decomposition, and QR factorization. Understand solution spaces for consistent and inconsistent systems. Explore practical applications including circuit analysis, engineering design, and optimization problems. Develop both hand-calculation and computational competencies.

• Gaussian elimination and row-reduction algorithms
• LU factorization and its computational advantages
• QR factorization and least-squares problems
• Consistency, existence, and uniqueness of solutions
• Applications to circuit analysis and engineering
• Numerical stability and computational considerations

Estimated time: 28 hours

Module 4: Linear Transformations and Matrix Representations

Explore linear transformations as the central objects of linear algebra. Understand how matrices represent transformations, and how changing bases changes matrix representations. Study geometric transformations including rotations, projections, and reflections. Connect abstract linear map properties to concrete matrix computations and visualizations.

• Definition and properties of linear transformations
• Kernel and image (range) of transformations
• Rank-nullity theorem and applications
• Matrix representations relative to different bases
• Change of basis and similarity
• Geometric transformations (rotations, reflections, projections)

Estimated time: 24 hours

Module 5: Eigenvalues, Eigenvectors, and Diagonalization

Master the theory and computation of eigenvalues and eigenvectors—critical for understanding long-term behavior of systems. Explore characteristic polynomials, diagonalization, and the spectral theorem. Study applications to dynamical systems, stability analysis, and iterative processes. Develop intuition for what eigenvectors reveal about matrix behavior.

• Characteristic polynomials and eigenvalue equations
• Computing eigenvalues and eigenvectors
• Diagonalization and similar matrices
• Spectral theorem for symmetric matrices
• Applications to dynamical systems and stability
• Power method and iterative eigenvalue computation

Estimated time: 28 hours

Module 6: Inner Products, Orthogonality, and Advanced Topics

Deepen understanding of geometric structure through inner products and orthogonal decompositions. Explore orthogonal matrices, orthogonal diagonalization, and the role of symmetry in linear algebra. Study singular value decomposition (SVD) and its applications to data analysis and machine learning. Connect elegant theory to modern computational applications.

• Inner products and induced norms
• Orthogonal and orthonormal bases
• Orthogonal matrices and unitary transformations
• Orthogonal diagonalization of symmetric matrices
• Singular Value Decomposition (SVD)
• Applications to principal component analysis and data compression

Estimated time: 22 hours

Module 7: Applications and Capstone Project

Synthesize linear algebra knowledge through real-world applications and a comprehensive capstone project. Apply matrix methods to machine learning (least-squares classification), computer graphics (3D transformations), quantum computing (state vectors), and engineering systems. Choose a project combining theoretical concepts with computational implementation, demonstrating mastery of the course material.

• Machine learning applications (linear regression, classification)
• Computer graphics and 3D transformations
• Quantum computing fundamentals via linear algebra
• Graph theory and network analysis
• Capstone project: Choose one major application domain and develop a complete solution using theory and code

Estimated time: 20 hours

Prerequisites

• Precalculus or equivalent (functions, basic trigonometry)
• Comfort with mathematical notation and abstract thinking
• Programming experience in Python, MATLAB, or equivalent (or willingness to learn basics)

What You'll Be Able to Do After

• Solve complex systems of linear equations using multiple methods and understand geometric interpretations
• Understand and compute eigenvalues/eigenvectors and apply spectral theory to real problems
• Represent linear transformations as matrices and understand how basis changes affect representations
• Apply linear algebra to machine learning, optimization, and engineering challenges
• Use computational tools (Python/MATLAB) to solve practical linear algebra problems at scale
• Write mathematical proofs and clearly communicate linear algebra concepts to others