Complete Curriculum for Linear Algebra for AI, Neural Networks, and Large Language Models (GPT/LLM) (日記)





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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