Linear Algebra
Fundamental concepts and applications in data science and machine learning
Why Linear Algebra?
Linear algebra forms the mathematical foundation for many machine learning algorithms and data science techniques. It provides the tools for handling high-dimensional data, solving systems of equations, and understanding transformations.
Key Concepts
Core Components
- Vectors and Vector Spaces
- Matrices and Operations
- Linear Transformations
- Eigenvalues and Eigenvectors
- Matrix Decompositions
Matrix Operations
Matrix Multiplication
C[i,j] = Σ(A[i,k] * B[k,j])Matrix Properties
- • Associative: (AB)C = A(BC)
- • Not Commutative: AB ≠ BA
- • Distributive: A(B+C) = AB + AC
- • Identity: AI = IA = A
Implementation with NumPy
import numpy as np
# Create matrices
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
# Matrix operations
matrix_sum = A + B
matrix_product = np.dot(A, B) # or A @ B
transpose = A.T
inverse = np.linalg.inv(A)
# Eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(A)
# Singular Value Decomposition
U, S, V = np.linalg.svd(A)
# Solve linear system Ax = b
b = np.array([1, 2])
x = np.linalg.solve(A, b)
print("Matrix Product:\n", matrix_product)
print("\nEigenvalues:\n", eigenvalues)
print("\nSolution to Ax = b:\n", x)Applications in ML
Principal Component Analysis
Dimensionality reduction using eigenvalue decomposition
Neural Networks
Matrix operations in neural network layers