Overview
Linear systems underpin many of the foundational algorithms in modern AI, optimization problems in blockchain, and quantum gate operations. This guide explores their mathematical properties and real-world applications.
Core Concepts
Matrix Operations
Matrices represent transformations and systems of equations. Fundamental for machine learning models and quantum state manipulations.
Eigenvalues
Analyze system stability and find principal components in dimensionality reduction for AI algorithms.
Applications
AI Neural Networks
Linear algebra forms the basis for weights and activations in deep learning models like CNNs.
Quantum Computing
Qubit states are represented using vectors and transformed via unitary matrices.
Blockchain Optimzation
Solving linear systems improves consensus algorithm efficiency in decentralized networks.
Code Examples
// Matrix Multiplication Example
const A = [[1, 2], [3, 4]];
const B = [[5, 6], [7, 8]];
function multiplyMatrices(a, b) {
const result = Array.from({length: a.length}, () => Array(b[0].length));
for (let i = 0; i < a.length; i++) {
for (let j = 0; j < b[0].length; j++) {
result[i][j] = 0;
for (let k = 0; k < b.length; k++) {
result[i][j] += a[i][k] * b[k][j];
}
}
}
return result;
}
console.log(multiplyMatrices(A, B));