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- Projection onto W with orthogonal basis {u₁, …, uₚ}: ŷ = Σ (y·uᵢ)/(uᵢ·uᵢ) uᵢ. Then z = y − ŷ ∈ W⊥, and ŷ is the closest point in W to y.
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- Gram–Schmidt: v₁ = x₁, v₂ = x₂ − (x₂·v₁)/(v₁·v₁) v₁, v₃ = x₃ − proj onto v₁ − proj onto v₂ … then normalize for an orthonormal basis.
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Apply Gram–Schmidt to x₁ = (3, 6, 0), x₂ = (1, 2, 2).
v₁ = (3, 6, 0). x₂·v₁ = 15, v₁·v₁ = 45, so v₂ = (1, 2, 2) − (1/3)(3, 6, 0) = (0, 0, 2). Orthogonal basis {(3, 6, 0), (0, 0, 2)}.
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- QR: A = QR, Q has orthonormal columns (Gram–Schmidt on A’s columns), R = QᵀA is upper triangular with positive diagonal.
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