Gram-Schmidt Orthogonalization Calculator
Enter one vector per line. Use commas or spaces between values.
What this Gram-Schmidt calculator does
The Gram-Schmidt process takes a set of vectors and transforms them into an orthogonal (or orthonormal) basis that spans the same subspace. This calculator automates that process and shows each projection step so you can learn the method, not just get the final answer.
- Accepts vectors in any dimension (as long as all vectors have the same length).
- Computes an orthogonal basis u1, u2, ....
- Optionally normalizes to produce an orthonormal basis q1, q2, ....
- Identifies linearly dependent vectors during processing.
How to use it
1) Enter your vectors
Put one vector per line. For example, in three dimensions you might enter: 1,1,0, 1,0,1, and 0,1,1.
2) Choose precision and normalization
Decimal precision controls how many digits are displayed. If you check “Also compute orthonormal basis,” each orthogonal vector is divided by its norm to produce unit vectors.
3) Click calculate
You’ll get a result summary, an orthogonal basis, optional orthonormal basis, and a detailed step-by-step breakdown with projection coefficients.
Gram-Schmidt process (quick explanation)
Given vectors v1, v2, ..., vk, Gram-Schmidt constructs:
- u1 = v1
- u2 = v2 - proju1(v2)
- u3 = v3 - proju1(v3) - proju2(v3)
where proju(v) = (v·u / u·u)u. If needed, normalize each ui to get unit vectors qi = ui / ||ui||.
Why this matters
Orthogonal and orthonormal bases are easier to work with numerically and conceptually. They show up in QR decomposition, least-squares fitting, signal processing, data science, and many machine learning workflows.
- Linear algebra classes: clean, step-by-step study support.
- Numerical methods: stable basis generation for matrix factorizations.
- Applied modeling: decorrelating features and interpreting geometric structure.
Common pitfalls
Dimension mismatch
Every vector must have the same number of components. If one row has a different length, the process is not defined.
Linear dependence
If a vector is completely explained by previous vectors, its orthogonal remainder is zero. The calculator reports this and continues with independent vectors.
Rounding effects
Tiny values near zero may appear because of floating-point arithmetic. Increase precision to inspect those values more closely.
FAQ
Does order matter?
Yes. Different input orders can produce different orthogonal bases, though they span the same subspace.
Can I use this for 2D, 3D, or higher dimensions?
Yes. The calculator works in any finite dimension supported by your input.
Is this the same as QR decomposition?
Gram-Schmidt is one way to build the Q factor in QR decomposition. If your vectors are matrix columns, the orthonormal vectors correspond to columns of Q.