Gram-Schmidt Orthogonalization

Orthonormal basis builder

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Vector Set

Orthonormal Basis

e1(0.7071, 0.7071, 0.0000)

e2(0.4082, -0.4082, 0.8165)

e3(-0.5774, 0.5774, 0.5774)

Steps

e1 = u1/||u1|| [||u||=1.4142]→ (0.7071, 0.7071, 0.0000)

u2 -= proj(e1,v2)·e1 [proj=0.7071]→ (0.5000, -0.5000, 1.0000)

e2 = u2/||u2|| [||u||=1.2247]→ (0.4082, -0.4082, 0.8165)

u3 -= proj(e1,v3)·e1 [proj=0.7071]→ (-0.5000, 0.5000, 1.0000)

u3 -= proj(e2,v3)·e2 [proj=0.4082]→ (-0.6667, 0.6667, 0.6667)

e3 = u3/||u3|| [||u||=1.1547]→ (-0.5774, 0.5774, 0.5774)

Verify: eᵢ·eⱼ

e1·e1 = 1.0000e1·e2 = 0.0000e1·e3 = 0.0000e2·e2 = 1.0000e2·e3 = -0.0000e3·e3 = 1.0000

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About Gram-Schmidt Orthogonalization

A Gram-Schmidt orthogonalization calculator. Transforms a set of linearly independent vectors into an orthonormal basis. Shows projection and subtraction steps. Verifies orthonormality. Select from preset vector sets. All calculations are client-side. Foundation for QR decomposition.

Gram-Schmidt Orthogonalization Features

Step-by-step
Projections
Normalization
Verify ortho
Presets

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