Gram-Schmidt Orthogonalization
Vector v₁ (space-separated)
Vector v₂
Vector v₃ (optional)
The Gram-Schmidt process converts a set of linearly independent vectors into an orthonormal basis — vectors that are mutually perpendicular and each have length 1. It is used in linear algebra, QR decomposition, and machine learning.
- 1Project each vector onto already-processed vectors
- 2Subtract those projections (orthogonalise)
- 3Divide result by its length (normalise)
- 4Projection: proj_u(v) = (v·u/u·u)u
v₁=(1,1,0), v₂=(1,0,1)=e₁=(0.707,0.707,0), e₂=(0.408,−0.408,0.816)
| Step | Operation | Result |
|---|---|---|
| 1 | Take v₁ as-is | u₁ = v₁ |
| 2 | Normalise | e₁ = u₁/|u₁| |
| 3 | Subtract projection | u₂ = v₂ − proj_e₁(v₂) |
| 4 | Normalise | e₂ = u₂/|u₂| |
References
🔒
100% Ücretsiz
Kayıt yok
✓
Hassas
Doğrulanmış formüller
⚡
Anında
Anında sonuçlar
📱
Mobil uyumlu
Tüm cihazlar