Arcadia AI — Master's Program
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Program  /  Semester 1 — AI Foundations
Course

Linear Algebra: Spaces, Embeddings, Transformations

The mathematical foundation: vector spaces, linear maps, and the geometry of data

Instructor to be announced

About the course

This course treats linear algebra as the language of modern machine learning. Students study vector spaces, eigendecompositions, and the singular value decomposition (SVD), with an emphasis on their role in embeddings, compression, and neural network transformations. The course builds the mathematical apparatus needed to understand transformers, PCA, and graph neural networks covered in later semesters.

What you'll learn

✓Apply SVD and spectral decomposition to dimensionality reduction and latent analysis
✓Explain the geometric meaning of matrix operations in neural network architectures
✓Compute gradients with respect to matrix parameters for optimization problems
✓Assess the numerical stability of linear algebra algorithms

Key topics

•Vector spaces and linear subspaces
•Linear maps and matrices
•Decompositions: LU, QR, spectral, SVD
•Projections, least squares, the Moore–Penrose pseudoinverse
•Embeddings and metrics in high-dimensional spaces
•Concentration of measure and the curse of dimensionality
•Randomized methods (Johnson–Lindenstrauss, random projections)
•Tensors and multilinear algebra
•Fundamentals of matrix differentiation
This description was generated automatically and has not yet been reviewed by an instructor — it's a draft for discussion.