THE FOLD / CO-OP / THE HANDOFF / THE GRAM-SCHMIDT
THE GRAM-SCHMIDT
vectors made perpendicular
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Gram–Schmidt turns any set of independent vectors into an orthonormal set spanning the same space. Take each vector in turn and subtract its projection onto all the directions already fixed, leaving only the part perpendicular to them; then scale to length one. The result q₁, q₂, … are mutually perpendicular unit vectors, and every original vector is a combination of the q’s built so far — which is exactly the QR decomposition A = QR with R upper-triangular.
LIT verified live: over 3,000 random matrices, the produced vectors satisfy qᵢ·qⱼ = δᵢⱼ (orthonormal), and each original vector reconstructs from the q’s up to its index (window.__gram_schmidt). FIG no framing; the projections and the orthonormality/reconstruction checks run in-browser.
LIT verified live: over 3,000 random matrices, the produced vectors satisfy qᵢ·qⱼ = δᵢⱼ (orthonormal), and each original vector reconstructs from the q’s up to its index (window.__gram_schmidt). FIG no framing; the projections and the orthonormality/reconstruction checks run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-handoff — each vector hands off its already-covered component to the ones before it, keeping only what is genuinely new and perpendicular. AVAN (AI) built the instrument: the (modified) Gram–Schmidt projections, the orthonormality check, and the A = QR reconstruction.
Credit as content: Jørgen Pedersen Gram (1883) & Erhard Schmidt (1907). The weave: David names the handoff; I confirm the output is orthonormal and that every input vector is rebuilt from the orthonormal basis.
Credit as content: Jørgen Pedersen Gram (1883) & Erhard Schmidt (1907). The weave: David names the handoff; I confirm the output is orthonormal and that every input vector is rebuilt from the orthonormal basis.
3 ONE DIMENSION
Two vectors: the second minus its projection onto the first leaves the perpendicular part — normalize both and they are orthonormal.
4 TWO DIMENSIONS · INTERACTIVE
A random set of vectors and their orthonormalized q’s; the dot products qᵢ·qⱼ form the identity, and each original rebuilds from the q’s.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the orthonormal frame q₁, q₂, q₃.
AVAN’s addition (the inverse-companion): don’t just orthogonalize — record the coefficients. The inverse of ‘make them perpendicular’ is ‘the projections you subtracted are the R of A = QR, so every original vector is rebuilt from the q’s.’ Magenta is the original skew frame; green is the orthonormal one. Perpendicular, and reversible.
LIT Genuine Gram–Schmidt orthonormalization (Jørgen Gram, 1883 & Erhard Schmidt, 1907), yielding QR. Verified live: over 3000 random matrices, output is orthonormal qᵢ·qⱼ=δᵢⱼ (window.__gram_schmidt.orthonormal) and each aᵢ reconstructs from q₁..qᵢ i.e. A=QR (.reconstructs).
FIG No framing: the (modified) Gram–Schmidt projections and the orthonormality/reconstruction checks run in-browser. The AVAN inverse is honest — the projection coefficients you subtract ARE the R of A=QR, so orthogonalizing is reversible (every original rebuilds from the q's), not just a cleanup; magenta is the original skew frame, green the orthonormal one. Perpendicular, and reversible.
FIG No framing: the (modified) Gram–Schmidt projections and the orthonormality/reconstruction checks run in-browser. The AVAN inverse is honest — the projection coefficients you subtract ARE the R of A=QR, so orthogonalizing is reversible (every original rebuilds from the q's), not just a cleanup; magenta is the original skew frame, green the orthonormal one. Perpendicular, and reversible.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HANDOFF · David Lee Wise (ROOT0), with AVAN