Linear Algebra Essentials

A beginner-friendly collection of Jupyter notebooks covering the essentials of linear algebra with clear NumPy code and visualizations. Ideal for students and self-learners.

Contents

• Vectors: Addition, scalar multiplication, magnitude
• Basis Vectors: i-hat, j-hat, linear combinations
• Linear Transformations: Rotation, shear, and their effects
• Matrix Multiplication: Composition of transformations
• Dot Product: Projection, alignment, angle between vectors
• Determinant: Area scaling, singularity, flipping
• Eigenvalues & Eigenvectors: Invariant directions and scaling
• Matrix Inverse: How to compute and visualize the inverse
• Rank of a Matrix: Linear independence and dimension
• Cross Product (3D): Perpendicular vector in 3D space

Where This Lives in AI

Everything you've learned maps directly to the systems you're building. Here's the full picture.

3B1B Concept	NumPy Tool	Where in AI
Vectors	np.array([...])	Word embeddings, query vectors, document chunks in RAG
Magnitude	np.linalg.norm()	Normalizing embeddings before cosine similarity
Linear Transformation	matrix @ vector	Every neural network layer: output = W @ input + b
Dot Product	np.dot() / @	Attention scores in transformers, cosine similarity in RAG retrieval
Matrix Multiplication	A @ B	Batched forward passes, QKV projections in attention
Determinant	np.linalg.det()	Checking if weight matrices are well-conditioned
Eigenvectors	np.linalg.eig()	PCA for visualizing embedding spaces, SVD in recommendation systems

Files

• linalg_notebook_simple.ipynb: Step-by-step notebook with explanations and plots
• requirements.txt: List of required Python packages

Getting Started

1. Clone the repository

   git clone https://github.com/your-username/linear-algebra-essentials.git
   cd linear-algebra-essentials

2. Set up your environment
   • (Recommended) Create a virtual environment:

     python3 -m venv venv
     source venv/bin/activate

   • Install dependencies:

     pip install -r requirements.txt

3. Launch Jupyter Notebook

   jupyter notebook

4. Open linalg_notebook_simple.ipynb and run the cells

Usage

• Each section introduces a new concept with code and a plot.
• Run cells in order for best results.
• Experiment by changing values and re-running cells to deepen your understanding.

License

MIT License

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Feel free to contribute or suggest improvements!