Inner product spaces
Lesson 10 of 10
The Gram–Schmidt process
Objective. Turn a basis into an orthonormal basis.
The key idea
Take each vector in turn, subtract its projections onto the vectors already processed, then normalise what remains.
Worked example
Orthogonalise v₁ = (1, 1), v₂ = (1, 0).
- 1u₁ = (1, 1).
- 2proj = ((1)/2)(1, 1) = (0.5, 0.5).
- 3u₂ = (1, 0) − (0.5, 0.5) = (0.5, −0.5).
Answer: u₁ = (1, 1), u₂ = (0.5, −0.5)
Practice
Each question comes with a picture and a listen button. Try it first, then reveal the answer.
Common mistake. Normalising before subtracting all the earlier projections.
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