What is the Gram-Schmidt process?

It is an algorithm that takes a set of linearly independent vectors and produces a set of orthogonal (or orthonormal) vectors that span the same subspace. Each new vector is obtained by subtracting the projections onto all previously computed orthogonal vectors.

What is the difference between orthogonal and orthonormal?

Orthogonal vectors are mutually perpendicular (their dot products are zero). Orthonormal vectors are additionally unit vectors — each has a magnitude (norm) of exactly 1. The orthonormal set is obtained by dividing each orthogonal vector by its norm.

What happens if I enter linearly dependent vectors?

If one vector is a linear combination of the others, the Gram-Schmidt process will produce a zero vector at that step. The calculator detects this and shows an error rather than outputting a meaningless result.

Does the order of input vectors matter?

Yes. The first vector is taken as-is. The second is orthogonalized against the first, and the third (if present) against both previous results. Changing the order produces a different (but equally valid) orthogonal basis.

Can I use this for 2D vectors?

Yes. Select "2D" and enter 2-component vectors. The calculator handles both 2D (2 vectors) and 3D (2 or 3 vectors) cases.

Gram-Schmidt Calculator

Enter 2 or 3 vectors to compute an orthogonal and orthonormal basis using the Gram-Schmidt process, with full step-by-step working.

Input Vectors

Dimension

Number of Vectors

Enter your vectors and click Calculate to see results.

Summary

The Gram-Schmidt process converts a set of linearly independent vectors into an orthogonal (or orthonormal) basis for the same vector space. This calculator supports 2 or 3 vectors in 2D or 3D, computes each orthogonal basis vector by subtracting projections onto previously computed vectors, and then normalizes each result to produce unit vectors. All intermediate steps are shown so you can follow the working exactly.

How it works

Select the dimension (2D or 3D) and the number of vectors (2 or 3).
Enter the components of each input vector.
Click "Calculate" to run the Gram-Schmidt process.
The tool computes each orthogonal vector by subtracting projections of previous basis vectors.
Each orthogonal vector is then divided by its norm to produce an orthonormal unit vector.
Step-by-step working is displayed so you can verify every intermediate value.

Use cases

Linear algebra homework and exam preparation.
Computing an orthonormal basis for QR decomposition.
Verifying manual Gram-Schmidt calculations.
Understanding vector projection and orthogonality concepts.
Machine learning feature orthogonalization.
Physics and engineering problems requiring orthonormal frames.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu