Turn a set of vectors into an orthonormal basis.
Last updated: June 2026 | By Patchworkr Team
Enter vectors as rows. The calculator produces an orthonormal basis using the classical Gram-Schmidt process.
It removes projections onto earlier vectors and then normalizes the result.
Gram-Schmidt is a building block for QR factorization and orthogonal coordinate systems.
Use the default three vectors.
1. Start with [1, 1, 0]
2. Remove its projection from the next vectors
3. Normalize each orthogonal vector
Final answer: an orthonormal basis
How many vectors can I add?
You can add as many as you need for the basis.
What if the vectors are dependent?
A dependent set can produce zero-length output vectors.
Is order important?
Yes. Gram-Schmidt depends on the input order.
Are the outputs normalized?
Yes. The resulting vectors have unit length when possible.
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