Change-of-Basis Review¶
Review procedure — instantiates Coherent Linear Space Design
Re-expresses the same vectors in a second basis and checks which conclusions survive the switch and which were artifacts of the old coordinates.
A single coordinate table tells you where everything sits in one basis — but it cannot tell you whether a conclusion is about the space or merely about that basis. The Change-of-Basis Review answers that question by re-expressing the same elements in a second, deliberately different basis and auditing what changes. Its defining move is the invariance test: it sorts every downstream conclusion into two piles — the basis-invariant claims that survive any legitimate change of frame (and are therefore about the underlying space) and the coordinate-dependent claims that shift or flip when the basis rotates (and were therefore artifacts of an arbitrary choice). Where the Basis & Coordinate Table fixes one frame and records it, this review moves between frames on purpose to separate substance from bookkeeping.
Example¶
A risk team holds a portfolio expressed in raw asset weights and has been telling stakeholders that "Asset X is the single biggest source of risk." The Change-of-Basis Review re-expresses the same portfolio in a factor basis — market, size, and value factors derived from the covariance structure. Two things happen. The claim about Asset X flips: in the factor frame the risk is dominated not by any one asset but by a broad market exposure, and no single asset stands out. Meanwhile, other quantities are untouched by the change of frame — the portfolio's total variance, and the fact that one dominant factor explains most of it, are the same numbers in both bases because they are similarity invariants of the covariance operator. The review's verdict: "Asset X is the biggest risk" was a coordinate artifact of the raw-weight basis, while "one factor dominates" is a real, basis-invariant feature. It also flags a caution — the factor axes themselves are not unique, because an oblique rotation of them fits the data equally well, so their individual labels must not be over-interpreted.[n1]
How it works¶
- Build the change-of-basis map — an invertible transform from the original basis to the target basis, and check it is well-conditioned.
- Transform the coordinates of every element and re-derive the conclusions in the new frame.
- Classify each conclusion as basis-invariant (unchanged) or coordinate-dependent (changed), anchoring on quantities known to be invariant — norms, trace, determinant, eigenvalues.
- Reconcile and report — retain only invariants as conclusions about the space; demote coordinate-dependent numbers to descriptive, frame-specific statements.
Tuning parameters¶
- Target basis — an orthogonal rotation, an oblique one, or a data-derived basis. A more aggressive change probes invariance harder but can be less interpretable.
- Invariance criterion — which quantities must be preserved for a conclusion to "count." Strict criteria (exact equality) versus tolerant ones (within noise).
- Transform conditioning — how near-singular the change-of-basis map may be; an ill-conditioned transform amplifies error and can corrupt the comparison.
- Re-derivation scope — spot-checking a few headline claims versus re-running the entire analysis in the new frame. Broader scope catches more artifacts at more cost.
When it helps, and when it misleads¶
Its strength is separating substance from coordinate artifact: it is the discipline that stops a team from shipping a conclusion that a rotation of the axes would erase, and it identifies the genuinely frame-independent features worth acting on.
Its failure mode is mistaking a merely convenient basis for a privileged one — treating the factor axes, or the principal components, as if they named real underlying causes when an equally valid rotation would tell a different story. That is the well-known factor-rotation indeterminacy: infinitely many rotations fit the same data, so the loadings of any one rotation are not unique.[n1] A second failure is an ill-conditioned transform silently corrupting the re-expression. The guarding discipline is to report only invariants as conclusions, treat all basis-dependent numbers as description rather than explanation, and check the transform's conditioning before trusting the comparison.
How it implements the components¶
The review fills the archetype's interpretation face — it governs what coordinates are allowed to mean once the basis is recognized as a choice:
interpretive_coordinate_semantics— its core: it decides which coordinate-level statements are meaningful (invariant) versus artifacts (frame-dependent).basis_dimension_coordinate_record— it operates on the recorded coordinates of two bases and the map between them, comparing the two records.linear_combination_admissibility_rule— the change-of-basis transform re-expresses each element as a linear combination in the new basis, and the review confirms the admissible combinations are preserved.
It does not declare the carrier or operations (the Vector-Space Specification Sheet), verify the axioms (the Linear-Axiom Verification Checklist), test a specific target's reachability (the Linear-Combination Membership Test), or register nonlinear exceptions (the Nonlinear-Boundary Stress Test).
Related¶
- Instantiates: Coherent Linear Space Design — the review enforces the archetype's warning that coordinates are basis-relative, not intrinsic.
- Consumes: Basis & Coordinate Table — it needs a recorded coordinate frame to transform from before it can compare frames.
- Sibling mechanisms: Vector-Space Specification Sheet · Linear-Axiom Verification Checklist · Basis & Coordinate Table · Linear-Combination Membership Test · Zero-Span Linearity Check · Linear Embedding Diagnostics · Nonlinear-Boundary Stress Test
Editorial Notes¶
Form Classification¶
Form family: Assessment, Review & Assurance
Rationale: Re-expresses the same vectors in a second basis and checks which conclusions survive the switch and which were artifacts of the old coordinates, making its operative form a bounded evaluation of existing evidence or work that produces a finding or disposition.
Independent corroboration: The frozen evidence defines Change-of-Basis Review as 'Re-expresses the same vectors in a second basis and checks which conclusions survive the switch and which were artifacts of the old coordinates', so its operative form is Assessment, Review & Assurance.
Review outcome: Independent reviewer agreement; high confidence.
Origin Attribution¶
Primary origin: Mathematics
Origin pattern: Cross-disciplinary synthesis
Present-day reach: Multi-domain
Rationale: Linear algebra supplies basis transformations and the distinction between invariant properties and coordinate artifacts.
Related originating lineages:
- Physics — Frame-change reasoning made invariance under alternative representations a practical test of physical substance.
Review resolution: Mathematics is the agreed primary lineage because the review re-expresses invariant objects under alternate coordinate bases. Physics contributes coordinate-invariance practice, and turning the mathematical operation into a robustness review is a multi-domain Encyclopedia synthesis.
Encyclopedia synthesis: The exact catalogued form synthesizes established practice rather than reproducing a single standard historical label.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
[n1] In factor analysis the factor loadings are unique only up to rotation: an orthogonal or oblique rotation (varimax, promax, and others) produces a mathematically equivalent fit with different loadings. This rotational indeterminacy is the canonical caution that a chosen basis is not a privileged one, and that per-axis interpretations must be made with care. ↩a ↩b