Gram-Schmidt without normalization

This tool applies the Gram-Schmidt procedure to a set of vectors in $\mathbb{R}^n$ and returns an orthogonal basis (not normalized) of the space they span.

It is the version to use when only orthogonality between the vectors matters, not their length: at every step the tool keeps the representative with reduced integer coordinates, avoiding square roots and decimal numbers at every stage of the computation.

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, and 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 orthogonal 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 directly an orthonormal basis (with unit-length versors) — for that there is the full Gram-Schmidt solver.

Step 1 — SET HOW MANY VECTORS AND OF WHICH DIMENSION

N° of vectors \(\left(k\right)\)\(\times\)Dimension \(\left(n\right)\)

Step 2 — SET THE VECTOR COORDINATES

RESET VALUESRANDOM VALUESQUICK RESULT

STEP-BY-STEP SOLUTION