independence

std.linalg.independence · Level L4

For k integer vectors in Rⁿ (the rows of V), by exact rank: the rank (the dimension of their span); whether they are linearly independent (rank k, only the zero combination gives 0); whether they span Rⁿ (rank n); and whether they are a basis, which needs both. Neither property stands in for the other.

independent ⇔ rank = k; spanning ⇔ rank = n

Signature

independence(V: f64[k, n]) → f64[], f64[], f64[], f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference exact rank, compared with k and n in exact rational arithmetic, on all 40 test cases.
All 160 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 39 of them exact, with zero error. Uncertified, too ill-conditioned at float32 for a finite bound: 1.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

rank_exact

Called by

—

sha256:dc1cd45fd5e6aa248b4be57aabc3f5614ab9239d0455349ddff5cb09b07cdc58

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.linalg.independence · Ω:independence
Pins: sha256:c79950aedac8e2186368a8b82e9e46fdedd1f603b6ad75c05c23f0ac888e3cb1this graph and the 2 it reaches through calls
Evidence: sha256:2713a83dc516057dfd72d16c492e5e6dbcb89338fd139256f54f6401b74eaf64the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.