📊 Linear Algebra Projects

👨‍🏫Under the Supervision of Prof. Morteza Haghir Chehreghani
🍂Spring 2021

This repository contains three projects focused on practical applications of linear algebra:

1. 🔢 Solving Linear Equations with LU Decomposition
2. 📉 Denoising Bitcoin Price Data Using Least Squares
3. 🖼️ Compressing Bitmap Images via SVD Decomposition

---

🧮 Project 1: Linear Equations Solver with LU Decomposition

📝 Overview

A Python implementation to solve systems of linear equations using LU decomposition and forward/backward substitution. This method efficiently handles multiple systems with the same coefficient matrix but different right-hand-side vectors.

⚙️ Requirements

• Python 3.x
• NumPy

📥 Input Format

The input consists of:

1. Two integers n (size of the square matrix A) and m (number of vectors b)
2. n lines representing the rows of matrix A
3. m lines, each containing a vector b

📋 Example Input:

3 5
5 6 2
4 5 2
2 4 8
18 7 2
4 5 8
15 7 6
11 9 5
13 12 12

📤 Output Format

Solutions for each b are printed up to 4 decimal places.

📊 Example Output:

75.0 -64.0 13.5
-14.0 13.0 -2.0
53.0 -45.0 10.0
0.5 1.5 -0.25
-10.0 11.0 -1.5

📉 Project 2: Denoising Bitcoin Price Data Using Least Squares

📝 Overview

Applies the Least Squares Method to smooth noisy Bitcoin price trends, revealing underlying patterns.

🎨 Simulated Result

🖼️ Project 3: Bitmap Image Compression via SVD Decomposition

📝 Overview

Compresses .bmp images using Singular Value Decomposition (SVD), reducing file size while preserving visual quality.

▶️ How to Run

1. Provide the path to a .bmp image
2. The script outputs a compressed version

🎨 Simulated Result

Original vs. Compressed Image

---

✨ Key Features

• 🧩 Modular Code Structure - Easy to understand and modify
• ⚡ Efficient Algorithms - Optimized implementations of core linear algebra operations
• 📚 Academic Foundation - Based on rigorous mathematical principles

🚀 Getting Started

1. Clone this repository
2. Install dependencies: pip install numpy matplotlib
3. Run individual project scripts

📜 License

This project is licensed under the MIT License - see the LICENSE file for details

🙏 Acknowledgments

Special thanks to Prof. Chehreghani for guidance on these implementations