Linear Algebra › Section F · Orthogonality and least squares › Topic F.4
The Gram-Schmidt process and QR
Orthogonal bases make projections and coordinates easy, but subspaces rarely come with one. The Gram-Schmidt process builds one from any basis, and recording its steps gives the QR factorisation, a workhorse of numerical computing.
In this topic
- From any basis to an orthogonal one
- The idea: remove what you already have
- The process in formulas
- Worked example: two vectors in R3
- Worked example: three vectors in R4
- The QR factorisation
- Computing R
- Worked example: a QR factorisation
- In practice: stability and history
- Common mistakes to avoid
- Key takeaways
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