QR Factorisation via Gram-Schmidt

Fact 1 of the Gram-Schmidt process note already previewed that Gram-Schmidt packages its own computation into a matrix factorisation. This note makes that factorisation precise and names it: $A=QR$, with $Q$'s columns orthonormal and $R$ upper triangular. This is the second great matrix factorisation of the two-part unit — Topic 2 of Linear Algebra I built $A=LU$ from elimination; this factors $A$ using orthogonality instead of elimination, and the payoff (numerically superior least-squares solving, previewed here and developed fully later in the topic) is exactly why both factorisations earn a place in a working quant's toolkit.

Learning objectives. By the end of this note you should be able to: