What is Vector Space Gram-Schmidt Calculator and Why it Matters?
Gram-Schmidt converts a linearly independent set into an orthonormal basis, simplifying projections and QR factorization.
Mathematical Formula and Theory
u₁ = v₁/‖v₁‖, then uⱼ = (vⱼ − Σ proj_{uᵢ} vⱼ) / ‖…‖. The resulting set is orthonormal.
How to Use This Calculator - Step by Step
- Enter one vector per line.
- Click Gram-Schmidt.
- Obtain the orthonormal set.
Solved Example with Full Calculation
v₁=[1,1,0], v₂=[1,0,1], v₃=[0,1,1] yields orthonormal basis u₁=[1,1,0]/√2, u₂=[1,−1,2]/√6, u₃=[−1,1,1]/√3.