Vector Search and Embeddings

  1. Mathematical Foundations: Vectors and Vector Spaces
    1. Understanding Vectors
      1. Definition of a Vector
        1. Vector Components and Magnitude
          1. Vector Direction and Orientation
            1. Representation in Coordinate Systems
              1. Vector Notation and Conventions
              2. Vector Spaces
                1. Definition of a Vector Space
                  1. Basis Vectors and Span
                    1. Dimensionality and Rank
                      1. Linear Independence
                        1. Subspaces and Orthogonality
                          1. Geometric Intuition in 2D and 3D
                          2. High-Dimensional Spaces
                            1. The Curse of Dimensionality
                              1. Effects on Distance Metrics
                                1. Impact on Search Algorithms
                                  1. Computational Complexity Growth
                                  2. Properties of High-Dimensional Geometry
                                    1. Sparsity of Data Points
                                      1. Concentration of Measure
                                        1. Volume and Surface Area Relationships
                                        2. Intuition and Visualization Challenges
                                        3. Essential Vector Operations
                                          1. Vector Addition and Subtraction
                                            1. Scalar Multiplication
                                              1. Dot Product and Inner Products
                                                1. Cross Product in 3D
                                                  1. Norms and Vector Length
                                                    1. L1 Norm (Manhattan)
                                                      1. L2 Norm (Euclidean)
                                                        1. Infinity Norm
                                                        2. Vector Normalization
                                                          1. Angle Between Vectors
                                                            1. Projection and Orthogonalization

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                                                          1. Introduction to Vector Search and Embeddings

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                                                          3. Data Representation: Embeddings