Linear Algebra Basics for Machine Learning: A Practical Guide to Vectors, Transformations, and SVD Decomposition

Linear algebra is the mathematical foundation of machine learning, but abstract symbols often deter beginners. This article introduces the GitHub open-source project 'linear-algebra-for-ml', designed for ML beginners. It implements core concepts like vector operations, linear transformations, and SVD using Python/NumPy, and combines visualizations to build an intuitive understanding, bridging theory and ML applications. Maintained by Zoha Khawaja, a software engineering student, the project aims to help learners with a programming background master linear algebra basics through a 'learning by doing' approach.

• Project Source: GitHub repository (link: https://github.com/zohaakhawajaa/linear-algebra-for-ml), released on June 6, 2026.
• Author: Zoha Khawaja (software engineering student, aspiring AI engineer).
• Positioning and Goals: As a learning resource and personal research project, its goals include: 1. Building an intuitive understanding of abstract concepts through examples and visualizations; 2. Hands-on implementation of core algorithms (not just theory); 3. Bridging linear algebra and ML applications.

Tech Stack: Python (concise syntax), NumPy (numerical computation), Matplotlib (visualization), Scikit-Learn (algorithm reference), Jupyter Notebook (interactive environment). Core Modules:

1. Vector Basics: Vector representation, addition/subtraction, scalar multiplication, dot product (similarity/orthogonality/projection), norms (applications in L1/L2 regularization).
2. Linear Transformations: Essence of matrix multiplication, shear transformations (applications in graphics/preprocessing), transformation visualization.
3. SVD Decomposition: Matrix decomposition concept, composition of U/Σ/V^T, implementation with NumPy/Scikit-Learn, matrix reconstruction and dimensionality reduction (associated with PCA).

The project's concepts are closely related to ML practice:

• Vectors: Word embeddings (Word2Vec/GloVe) use vectors to represent words, and dot product reflects semantic similarity.
• Linear Transformations: Neural network layers are essentially linear transformations plus non-linear activations; understanding transformations helps master feature representation.
• SVD: Collaborative filtering recommendation systems use matrix decomposition to fill ratings; PCA is an application of SVD (dimensionality reduction/denoising).
• Norms: L1/L2 regularization corresponds to L1/L2 norms, preventing model overfitting.

Learning Path:

1. Vector Basics (1-2 weeks): Implement operations, visualization, dot product/norm calculations.
2. Linear Transformations (2-3 weeks): Matrix multiplication, transformation visualization, shear transformations, exploring matrix features.
3. SVD (2-3 weeks): Manual calculation (small scale), implementation with NumPy/real data, image compression, understanding the relationship with PCA. Practical Recommendations: Hands-on implementation (rewrite code independently), visualization exploration (modify parameters to observe changes), apply real data, compare math with code, learn by teaching (explain concepts to others).

This project demonstrates an efficient learning path: starting from programming practice, building intuition through visualization, and gradually deepening into theory. Its value lies in: lowering the learning threshold (code replaces abstract symbols), building geometric intuition, bridging theory and application, and progressive content design. For self-learners and learners with a computer background, it is an excellent starting point to master the mathematical foundations of ML. Linear algebra is not an obstacle to ML, but a key to understanding algorithm principles—through such practical projects, more learners can cross the threshold and truly understand the mathematical logic behind algorithms.