How does the Gram-Schmidt Calculator work?

Project each vector onto already-processed vectors Then: Subtract those projections (orthogonalise) Then: Divide result by its length (normalise) Then: Projection: proj_u(v) = (v·u/u·u)u.

Can you give an example of how to use the Gram-Schmidt Calculator?

Example: Input v₁=(1,1,0), v₂=(1,0,1) gives a result of e₁=(0.707,0.707,0), e₂=(0.408,−0.408,0.816).

Is the Gram-Schmidt Calculator free to use?

Yes — completely free with no registration, download, or subscription required. All calculations happen instantly in your browser.

How accurate is the Gram-Schmidt Calculator?

Our Gram-Schmidt Calculator uses verified mathematical formulas and is accurate to multiple decimal places. Results are calculated in real-time using the same methods used by professionals.

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Gram-Schmidt Calculator

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.

Symbol	Name	Unit	Description
v₁, v₂, ..., vₙ	original linearly independent vectors	—	—
u₁, u₂, ..., uₙ	orthogonal vectors	—	—
e₁, e₂, ..., eₙ	orthonormal vectors (unit length)	—	—

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

›Gram-Schmidt — Khan Academy

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