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Gram-Schmidt calculator (orthonormal basis)

Turn a set of vectors into an orthonormal basis with the Gram-Schmidt process: each vector is made orthogonal to the previous ones and then normalized to unit length. Enter one vector per row on an editable grid and it returns the orthonormal basis, dropping any linearly dependent vectors. Orthonormal bases are the foundation of QR factorization, Fourier analysis and projections.

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Open Gram-Schmidt calculator (orthonormal basis) and you get an answer straight away, with no account to create. It covers each vector is made orthogonal to the previous ones and then normalized to unit length. Enter one vector per row on an editable grid and it returns the orthonormal basis, dropping any linearly dependent vectors. Orthonormal bases are the foundation of QR factorization, Fourier analysis and projections — adjust any of them and the result follows immediately.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What does Gram-Schmidt calculator (orthonormal basis) do?

Turn a set of vectors into an orthonormal basis with the Gram-Schmidt process: each vector is made orthogonal to the previous ones and then normalized to unit length. Enter one vector per row on an editable grid and it returns the orthonormal basis, dropping any linearly dependent vectors. Orthonormal bases are the foundation of QR factorization, Fourier analysis and projections.

What does a concrete case look like?

(1, 1), (1, 0) → (1, 1)/√2, (1, −1)/√2 — the tool shows every step in between, not just the final figure.

What does it take into account?

It factors in each vector is made orthogonal to the previous ones and then normalized to unit length. Enter one vector per row on an editable grid and it returns the orthonormal basis, dropping any linearly dependent vectors. Orthonormal bases are the foundation of QR factorization, Fourier analysis and projections. Change any of them and the output follows immediately.

When would I actually use this?

Checking a decomposition done by hand, solving a linear system, and finding the eigenvalues that describe how a transformation stretches space.

What is the most common mistake?

Inverting a matrix to solve a system. Elimination or a factorisation is both faster and far more stable numerically; an explicit inverse amplifies rounding error, especially when the matrix is close to singular.

How is Gram-Schmidt calculator (orthonormal basis) different from Column Space Calculator (basis of Col(A), rank, pivot columns)?

They sit next to each other but answer different questions: Column Space Calculator (basis of Col(A), rank, pivot columns) is the one to open when you need it to a basis of the column space of any matrix up to 6×6, taken from the ORIGINAL pivot columns rather than the reduced ones — with the rank, the free columns, the full RREF and every row operation. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Null Space Calculator (basis of Nul(A), nullity, parametric solution) is the closest one after this: Solve Ax = 0 for any matrix up to 8×8: a basis of the kernel in exact fractions and in cleared integers, the free variables, the parametric solution, the rank–nullity check and A·v = 0 verified exactly.

Where do the figures come from?

The decompositions follow their textbook definitions with partial pivoting where it applies. Results are exact for small integer matrices and subject to floating-point rounding otherwise.

Further reading

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