Introduction to Matrices

Linear Algebra for Machine Learning · Axiom Academy

The Language of Transformations A matrix is a rectangular array of numbers arranged in rows and columns. It's one of the most fundamental tools in linear algebra, used to represent data, transformations, and systems of equations. Think of a matrix as a compact way to organize and manipulate data. Instead of writing out 100 individual equations, you can represent them all at once using a single matrix equation! Watch what happens when we apply a matrix to a grid. The animation below shows how a matrix stretches, rotates, and reshapes geometric objects: Rotation, scaling, filters, and effects are all matrix operations applied to pixel coordinates and color channels. Game engines use transformation matrices to position objects in 3D space, rotate characters, and project onto screens. Neural networks process data through millions of matrix multiplications. Each layer applies a matrix transformation. Stress, strain, and force interactions in materials are represented and solved using matrix equations. Social networks and connectivity graphs are represented as adjacency matrices for analysis. Input-output models, supply chains, and market equilibrium are analyzed using matrix methods. Matrix Notation & Operations: How to write, add, multiply, and transpose matrices Linear Transformations: How matrices describe geometric transformations Matrix Properties: Special types like identity, diagonal, orthogonal, and invertible matrices

This is the written version of the interactive lesson above. See the full Linear Algebra for Machine Learning course.