Linear Algebra Visualizer (2026)

An Interactive Visualization Platform for Matrix Transformations, SVD, PCA, and Least Squares (LSE)

Linear algebra underlies modern machine learning, computer vision, medical imaging, and scientific computing. However, students often experience matrices as abstract symbolic procedures rather than geometric transformations.

Linear Algebra Visualizer is an interactive Streamlit based visualization platform designed to bridge algebraic computation and geometric intuition. Instead of treating matrices as static arrays of numbers, this system demonstrates how matrices act as geometric operators that rotate, scale, project, compress, and transform data in real time.

🚀 Live Demo

👉 Open Interactive Linear Algebra Visualizer
📎 Short Link

🔗 Project Access

• GitHub Repository
• Project Homepage (GitHub Pages)

Features

2D Linear Transformations

• Real time visualization of matrix actions on vectors and grids
• Rotation, scaling, shear, and custom matrices

3D Transformations

• Interactive 3D geometric transformations
• Visualization of basis changes and spatial mapping

Projection and Lifting

• 2×3 projection (dimension reduction)
• 3×2 lifting (dimension expansion)
• Geometric interpretation of linear mappings between spaces

Singular Value Decomposition (SVD)

• Visual breakdown of a matrix transformation into:
  • Rotation (Vᵀ)
  • Scaling (Σ)
  • Rotation (U)
• Interactive control of singular values
• Demonstration of rank and conditioning

Principal Component Analysis (PCA)

• Variance visualization
• Major and minor axis interpretation
• Covariance geometry explanation

Least Squares Estimation (LSE)

• Interactive demo of fitting an overdetermined system Ax ≈ b
• Visual intuition for noisy data and approximation
• Geometric meaning as projection:
  • b projected onto column space of A
  • residual r = b − Ax orthogonal to column space
• Optional concepts:
  • AᵀA x = Aᵀ b
  • connection to pseudoinverse and SVD

Image Compression

• SVD based image compression
• PCA based image compression
• Rank reduction demonstrations
• Visual comparison of compression levels

Educational Goals

This platform is designed to help students:

• Develop geometric intuition for matrix operations
• Understand the meaning of singular value decomposition (SVD)
• Visualize rank deficiency and ill conditioning
• Connect PCA to variance and covariance structure
• Understand least squares as projection and error minimization
• See dimensionality reduction and fitting as geometry

The goal is not merely to compute linear algebra but to make it visible.

Why This Project Matters

Modern AI systems rely heavily on linear algebra. From neural networks to medical imaging reconstruction, matrix operations drive nearly every computation. Yet many students struggle to build intuitive understanding of these concepts.

By integrating 2D and 3D transformations, SVD, PCA, least squares estimation, and image compression into a unified interactive environment, this project transforms linear algebra from symbolic manipulation into geometric experience.

This work further extends into immersive VR environments, enabling users to experience linear algebra concepts spatially rather than symbolically.

Technologies Used

• Python
• Streamlit
• NumPy
• Matplotlib or Plotly
• PIL (Image Processing)
• Manim (Mathematical Animation Engine, originally developed by 3Blue1Brown)

Project Structure

Visualizer2026/
│
├── Home.py              # Main application entry
├── pages/               # Multi-page interactive modules
├── assets/              # Images and visual assets
├── requirements.txt     # Dependencies
├── README.md            # Repository overview
└── index.md             # GitHub Pages homepage

🥽 VR Extension: Linear Algebra in Immersive Space

Experience linear algebra beyond the screen. The Linear Algebra Visualizer VR Project extends the platform into an interactive WebXR environment, allowing users to explore vectors, transformations, and geometric structures in 3D immersive space.

Instead of observing transformations, users can walk through them, manipulate objects directly, and develop spatial intuition in a fully interactive environment.

🔗 Access the VR Project

• 🧠 Linear Algebra Visualizer VR Project (Info Page)
• 🚀 Launch VR Live Demo (WebXR)

✨ Key Highlights

• Real-time 3D vector and matrix interaction
• Immersive exploration of transformations
• WebXR support (Meta Quest / VR browsers)
• Intuitive spatial understanding of abstract concepts

💡 This VR module represents the next step toward making mathematics fully experiential and interactive.

Future Development

• Educational lesson integration
• Additional machine learning algorithm demonstrations

Author

Brayden Miao
BM Studio · High School Research Project (2026)

License

This project is open for educational use.