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This introductory lecture covers the fundamentals of linear algebra within COMP1321 Mathematics II. You'll learn about vectors, their operations (addition and scalar multiplication), and how to form linear combinations. These concepts are the foundation for everything that follows — matrices, linear maps, and solving systems of equations. Lecturer: Dr Markus Brede.

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📋 At a Glance

• [ ] Understand what linear algebra is and why it matters for CS
• [ ] Know the difference between column vectors and row vectors
• [ ] Be able to convert between row and column vectors using transposition
• [ ] Perform vector addition (coordinate-wise)
• [ ] Perform scalar multiplication and understand its geometric effects
• [ ] Understand the geometric representation of vectors as arrows from the origin
• [ ] Know the properties of vector operations (commutativity, associativity, distributivity)
• [ ] Form linear combinations of vectors ($cv + dw$)
• [ ] Understand the geometric meaning of linear combinations (line, plane, 3D space)
• [ ] Express a vector as a linear combination and connect to systems of equations

1. Why Linear Algebra?

Linear algebra is "the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces through matrices."

Applications in Computer Science

• Machine Learning & NLP — Words are embedded as vectors in high-dimensional spaces. Similar words sit close together. Machine translation becomes finding a linear map (matrix) between two language vector spaces
• Recommender Systems — Amazon, Netflix, dating apps all use linear algebra under the hood
• Search Engines — Google's PageRank is fundamentally a linear algebra computation