Gram-Schmidt

This tool applies the Gram-Schmidt procedure to a set of vectors in $\mathbb{R}^n$ and returns an orthonormal basis of the space they span.

Enter the vectors as columns (integers or fractions, e.g. 1/2, -3), choose how many there are ($k$) and which space they live in ($n$), then press SYNTHETIC RESULT for just the final basis, or STEP-BY-STEP SOLUTION to see every intermediate calculation: projections, orthogonal parts, squared norms and versors, with the common factor highlighted whenever a simpler-coordinate representative is chosen.

If one of the vectors turns out to be linearly dependent on the previous ones, the tool flags it and discards it automatically: the final orthonormal basis will then have fewer than $k$ elements.

Do not use it for: sets with more than 8 vectors or in dimension higher than 8 (the tool’s limit), nor to get a purely orthogonal basis (without normalization) — for that there is the dedicated solver, which is faster because it avoids radicals.

Step 1 — SET HOW MANY VECTORS AND OF WHICH DIMENSION

Vectors \(\left(k\right)\)\(\times\)Dimension \(\left(n\right)\)

Step 2 — SET THE VECTOR COORDINATES

RESET VALUESRANDOM VALUESQUICK RESULT

STEP-BY-STEP SOLUTION