Linear Algebra for AI: From Fundamentals to Advanced Applications

Fundamental Operations:

• Vector Operations: Addition, dot product, norms
• Matrix Multiplication: Einstein summation convention
• Special Matrices: Identity, diagonal, symmetric
• Matrix Decomposition: LU, QR, Cholesky

Python Implementation:

import numpy as np

# Vector operations
v = np.array([1, 2, 3])
w = np.array([4, 5, 6])
dot_product = np.dot(v, w)  # 32

# Matrix operations
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
matmul = A @ B  # [[19, 22], [43, 50]]

AI Connection:

Neural network forward passes are essentially sequences of matrix multiplications with non-linear activations

Key Concepts:

• Eigenvalues/Eigenvectors: Ax = λx
• SVD: A = UΣVᵀ (generalization of eigen decomposition)
• PCA: Dimensionality reduction using SVD
• Low-Rank Approximations: Compression technique

Numerical Implementation:

# Eigen decomposition
eigenvalues, eigenvectors = np.linalg.eig(A)

# SVD in Python
U, S, Vt = np.linalg.svd(X)

# Truncated SVD (for dimensionality reduction)
k = 50  # Top 50 components
X_reduced = U[:, :k] @ np.diag(S[:k]) @ Vt[:k, :]
        

AI Applications:

Used in recommendation systems (collaborative filtering), image compression, and latent semantic analysis

Key Techniques:

• Gradients: ∇f(x) for vector x
• Jacobians: ∂y/∂x for vector-valued functions
• Hessians: Second derivatives for optimization
• Chain Rule: Fundamental to backpropagation

Backpropagation Example:

Operation	Forward Pass	Backward Pass
Linear Layer	z = Wx + b	∂L/∂W = ∂L/∂z · xᵀ
ReLU	a = max(0,z)	∂L/∂z = ∂L/∂a ⊙ (z > 0)

Performance Insight:

Efficient matrix calculus implementations enable training of models with billions of parameters