Part 1: Linear Algebra Foundations for Neural Networks

1.1 Vector and Matrix Representations
• Data Representation: Input vectors and feature vectors (e.g., images, text, audio)
• Model Representation: Weight matrices and bias vectors

1.2 Meaning and Application of Matrix Operations
• Linear Transformation: Input × Weight = Extracted Features
• Batch Processing: Efficient computations via matrix operations

1.3 Linear Mapping and Feature Extraction
• Fully Connected Layer: Mathematical model
• Interpretation: How weight matrices transform information

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Part 2: Linear Algebra for Gradient Computation and Optimization

2.1 Jacobian and Hessian Matrices
• Matrix Representation: For multivariable function derivatives
• Understanding Gradients: First and second-order derivative analysis

2.2 Gradient Descent Method
• Weight Update Rule:
W \leftarrow W - \eta \nabla L(W)
(η: learning rate, ∇L: gradient of the loss function)

2.3 Backpropagation via Matrix Operations
• Efficient Updates: Weight and bias adjustments
• Full-layer Gradient Propagation: Mathematical formulation

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Part 3: Understanding Neural Network Internal Structures

3.1 Eigenvalues, Eigenvectors, and Stability
• Analyzing Gradient Vanishing/Explosion
• Theoretical Stabilization of Model Behavior

3.2 Singular Value Decomposition (SVD) and Dimensionality Reduction
• Model Compression: Parameter reduction
• Principal Component Analysis (PCA): Feature extraction

3.3 Orthogonality and Regularization Techniques
• Weight Orthogonalization
• Mathematical Basis for Batch Normalization and Dropout

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Part 4: High-Dimensional Space and Semantic Vector Operations

4.1 High-Dimensional Space and the Curse of Dimensionality
• Dot Product, Cosine Similarity, Distance Calculations
• Semantic Distance Evaluation Between Vectors

4.2 Embedding Spaces and Matrix Factorization
• Word Embeddings: Word2Vec, GloVe, FastText
• Matrix Factorization: Discovering latent semantic spaces

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Part 5: Mathematical Structure of Large Language Models (GPT/LLM)

5.1 Vector Space Semantics
• Semantic Representation: Words, sentences, and documents as vectors
• Semantic Operations:
Example: king - man + woman ≈ queen
• Similarity Measurement: Cosine similarity and dot product

5.2 Transformer Architecture and Linear Algebra
• Generating Query, Key, and Value Matrices
• Self-Attention Mechanism:
\text{Attention}(Q, K, V) = \text{Softmax}(QK^T / \sqrt{d})V
• Multi-Head Attention: Capturing diverse features in parallel

5.3 Positional Encoding as Matrix Representation
• Adding Positional Information: Using sinusoidal functions

5.4 Multi-Layer Structures and Gradient Optimization
• Layer-wise Weight and Bias Design
• Optimizing Learning via Backpropagation

5.5 Model Compression and Memory Optimization
• SVD, Low-Rank Approximation, Distillation, Quantization
• Acceleration with CUDA, cuBLAS, MKL

5.6 Semantic Understanding and Text Generation Applications
• Next-Token Prediction and Sentence Generation
• Long-Range Dependency Modeling for Contextual Understanding

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Learning Objectives
• Explain the structure and operation of neural networks, transformers, and LLMs using linear algebra.
• Implement and apply mathematical models using Python and machine learning libraries.
• Bridge theory and practice in AI technology for both professional and academic applications.

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