COMP 2307 · Data Science: Linear Algebra

Led by Reimannian Topology Simulacrum

Vectors, matrices, tensors, and the geometry of feature space — the linear algebra that underlies every ML algorithm.

1. Module 1

   Scalars, Vectors, and the Geometry of Data

   Led by Reimannian Topology Simulacrum

   The question

   Why is cosine similarity preferred over Euclidean distance for comparing text documents? The answer is geometric: cosine measures angle, distance measures magnitude. When does each matter?

   Outcome

   The student can compute dot products, Euclidean distances, and cosine similarities.

   Sub-units

   1. ○ 1.1 Distance and Similarity Open this unit → ✓ ↻
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2. Module 2

   Matrices: Data as Rectangular Arrays

   Led by Reimannian Topology Simulacrum

   The question

   Linear regression's normal equation is β = (X^TX)^{-1}X^Ty — a matrix inverse. Why does inverting X^TX give the least-squares solution, and what breaks when the matrix is ill-conditioned?

   Outcome

   The student can perform matrix operations in NumPy and explain the normal equation.

   Sub-units

   1. ○ 2.1 The Normal Equation Open this unit → ✓ ↻
   2. ○ 2.2 Matrix Multiplication Errors Open this unit → ✓ ↻
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3. Module 3

   Tensors: The General Case

   Led by Reimannian Topology Simulacrum

   The question

   An image is a rank-3 tensor (H, W, 3). A batch of 100 images is rank-4 (100, H, W, 3). Every deep learning framework represents data as tensors. Why — and what operations do tensors support that matrices do not?

   Outcome

   The student can manipulate tensor shapes in NumPy and explain why deep learning uses tensors.

   Sub-units

   1. ○ 3.1 Work with Tensors Open this unit → ✓ ↻
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4. Module 4

   Linear Algebra and Machine Learning

   Led by Reimannian Topology Simulacrum

   The question

   The principal components of a dataset are the eigenvectors of the covariance matrix, ordered by eigenvalue. What does an eigenvalue represent, and why is the top eigenvector the direction of maximum variance?

   Outcome

   The student can compute a covariance matrix, find eigenvectors, and connect them to PCA.

   Sub-units

   1. ○ 4.1 Covariance and PCA Open this unit → ✓ ↻
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5. Module 5

   Why Linear Algebra Matters in Practice

   Led by Reimannian Topology Simulacrum

   The question

   When you scale features, you change the geometry — distances change, cluster shapes change, regression coefficients change. Choose three algorithms and describe the specific geometric operation each performs.

   Outcome

   The student can connect linear algebra to specific ML algorithms and explain what changing the feature space changes.

   Sub-units

   1. ○ 5.1 Final Essay: Geometry and Data Open this unit → ✓ ↻
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