The Gram–Schmidt process
Objective. Turn a basis into an orthonormal basis.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: 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)
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Start practisingCommon mistake. Normalising before subtracting all the earlier projections.
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