The Gram–Schmidt Process — Turning Messy Directions Into a Perfect Coordinate System

The Gram–Schmidt Process — Turning Messy Directions Into a Perfect Coordinate System Imagine you are trying to navigate through a new city. If the roads are at strange angles, weaving around unpredictably, it’s hard to get your bearings. But if the streets are arranged in neat, perpendicular directions—one north–south, one east–west—the city becomes easy to understand. You can describe any location using clean coordinates, and every direction feels organized. The Gram–Schmidt process is a method in linear algebra that does exactly this for vectors. It takes a messy, tilted, overlapping collection of directions and transforms them into a beautifully structured, perfectly perpendicular set. In math, we call this an orthonormal basis—a set of directions that are perpendicular to each other and each have unit length.