Gram-Schmidt and QR Factorization

Imagine you have a bunch of arrows pointing in all different directions. If you could replace them with arrows that are perpendicular and exactly one unit long, almost every linear algebra problem—solving equations, fitting data—gets much easier. This chapter shows you a simple, step‑by‑step method, called Gram‑Schmidt, that does exactly that: it turns any set of independent vectors into a neat, orthonormal set. And that same method gives us one of the most useful ways to break a matrix into pieces: the QR factorization .