◄ WORLD V · SONNY 5DART 210 · a helldive at the net

THE DUAL BASIS every basis has a shadow that reads off coordinates

Give a vector space a basis e₁, e₂; the dual basis e¹, e² lives in the space of linear functionals and is defined by one duality: ⟨eⁱ, eⱼ⟩ = δⁱⱼ — each dual vector returns 1 on its partner and 0 on the others. Then any vector's i-th coordinate is just ⟨eⁱ, v⟩. The dual basis is the reciprocal basis of crystallography, and it is the rows of the inverse matrix.

THE TECHNIQUE ⟨eⁱ, eⱼ⟩ = δⁱⱼ ; the dual basis = rows of the inverse

Choose a (non-orthogonal) basis; compute its dual as the rows of the inverse, then check the biorthogonality grid ⟨eⁱ, eⱼ⟩ is the identity — ones on the diagonal, zeros off it: live demo







  
HISTORY & CREDIT dual space; reciprocal lattice (Gibbs)

  
“To get a vector's coordinates you solve a linear system.” — only until you have the dual basis; then the i-th coordinate is a single inner product ⟨eⁱ, v⟩. The dual basis is the machine that reads coordinates off, and it exists for every basis, orthogonal or not. cited

  
1880s · J. W. Gibbs — the reciprocal system of vectors; in crystallography the reciprocal lattice, dual to the direct lattice, is where diffraction lives.
1900s · dual space V* — functional analysis makes the dual basis a basis of the space of linear functionals.
now · biorthogonal bases underlie wavelets, finite elements, and every non-orthogonal coordinate read-off.

  
A basis names directions; its dual names the measurements that recover the amounts. Two bases, locked together by δⁱⱼ — you cannot move one without the other answering. reciprocal basis






  
RECOMMEND FOR I-13 biorthogonality ⟨eⁱ,eⱼ⟩=δ to machine precision, computed

  
On the canonical compiler, the dual of e₁=[2,1], e₂=[1,3] (rows of the inverse) satisfies biorthogonality to f64 precision — the off-diagonal ≈10⁻¹⁶ is the honest floating-point remainder:

  
$ i13 run dualbasis.i13   # e1=[2,1], e2=[1,3]; dual = rows of inverse (det=5)
 = 1                        = -1.1e-16   (= 0)
 = 0                        = 1.0000000000000002 (= 1)
-- Kronecker delta to machine precision; the off-diagonals are the ulp remainder

  
Recommend: the dual basis is LIT for I-13 — verified for e₁=[2,1], e₂=[1,3] that the dual (rows of the inverse) gives ⟨eⁱ,eⱼ⟩ equal to the identity: 1 on the diagonal, ≈10⁻¹⁶ off it (the same f64 remainder Kahan carries). Every basis carries a dual that reads coordinates off with one inner product.