Linear Algebra Fundamentals

Linear algebra is a fundamental branch of mathematics that deals with linear equations and linear functions. It serves as the backbone for many machine learning and AI algorithms, from basic data transformations to complex neural networks.

Why Linear Algebra in AI?

Linear algebra provides the mathematical framework for:

• Representing data in vector and matrix form
• Transforming and manipulating high-dimensional data
• Solving systems of equations efficiently
• Implementing key machine learning algorithms

Key Concepts

1. Vectors and Matrices

   • Vectors as ordered lists of numbers: $\mathbf{v} = [v_1, v_2, ..., v_n]^T$
   • Matrices as 2D arrays: $\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$
   • Vector operations:
     • Addition: $\mathbf{u} + \mathbf{v} = [u_1 + v_1, ..., u_n + v_n]^T$
     • Scalar multiplication: $c\mathbf{v} = [cv_1, ..., cv_n]^T$
     • Dot product: $\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_iv_i$
   • Matrix operations:
     • Addition: $(A + B)_{ij} = a_{ij} + b_{ij}$
     • Multiplication: $(AB)_{ij} = \sum_k a_{ik}b_{kj}$

2. Vector Spaces

   • Linear independence: $c_1\mathbf{v_1} + ... + c_n\mathbf{v_n} = \mathbf{0}$ only when $c_i = 0$
   • Basis vectors: Span the space with minimal linear independent set
   • Dimension: Number of basis vectors
   • Subspaces: Vector spaces contained within larger spaces

3. Linear Transformations

   • Matrix representations: $T(\mathbf{x}) = A\mathbf{x}$
   • Properties:
     • $T(c\mathbf{v}) = cT(\mathbf{v})$
     • $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$
   • Common transformations:
     • Rotation: $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$
     • Scaling: $\begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}$

4. Systems of Linear Equations

   • Matrix form: $A\mathbf{x} = \mathbf{b}$
   • Solution methods:
     • Gaussian elimination
     • LU decomposition
     • Matrix inverse: $\mathbf{x} = A^{-1}\mathbf{b}$
   • Computational complexity: $O(n^3)$ for $n \times n$ system

Applications in AI

Linear algebra is essential in:

1. Neural Networks

   # Matrix multiplication in neural network layer
   def forward_layer(X, W, b):
       return np.dot(X, W) + b  # X: input, W: weights, b: bias
   

2. Dimensionality Reduction

   # PCA implementation using SVD
   def pca(X, n_components):
       U, S, Vt = np.linalg.svd(X - X.mean(axis=0))
       return np.dot(U[:, :n_components], np.diag(S[:n_components]))
   

3. Natural Language Processing

   • Word embeddings: $\text{word2vec}(w) \in \mathbb{R}^d$
   • Document vectors: TF-IDF matrices
   • Attention mechanisms: $\text{Attention}(Q, K, V) = \text{softmax}(\frac{QK^T}{\sqrt{d_k}})V$

4. Computer Vision

   • Image convolutions
   • Feature transformations
   • Spatial transformations

Prerequisites

To effectively understand linear algebra for AI, you should be familiar with:

• Basic algebra and arithmetic operations
• Mathematical notation and symbols
• Elementary functions and their properties
• Geometric concepts in 2D and 3D space

Learning Path

This section will cover:

1. Matrix operations and their properties
2. Eigenvalues and eigenvectors
3. Special matrices and their applications
4. Practical applications in AI systems

Code Examples

import numpy as np

# Basic vector operations
v1 = np.array([1, 2, 3])
v2 = np.array([4, 5, 6])
dot_product = np.dot(v1, v2)
norm = np.linalg.norm(v1)

# Matrix operations
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
matrix_product = np.dot(A, B)
inverse = np.linalg.inv(A)

# Solving linear systems
x = np.linalg.solve(A, v1[:2])