projection concept
| Definition | The component of a vector along a unit direction. Equation 0.2. |
|---|---|
| Example | The projection of (3, 4) onto the unit direction (1, 0) is (3, 0), and the residual (0, 4) is orthogonal to it. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id projection, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 0.2. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/Projection.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 0.2.
\[\operatorname{proj}_u(x)=(u\cdot x)\,u,\qquad \|u\|=1.\]
Conditions
- The component of a vector along a unit direction, the direction scaled by their dot product. Projecting twice is projecting once, the remainder is orthogonal to the direction, the projection is no longer than the vector, and adding anything orthogonal to the direction leaves it unchanged.
- The last is the quotient a one-direction reader takes. Row normalization is the projection onto the read subspace of the geodesic-rank consumer, and the invariant-nuisance split of the ledger is the same statement for a general consumer.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.2 of Data Mining as Observation, with
the program’s angular projection in Volume 14 chapter 9,
geometric-observation/chapters/ch09_legibility.md:31-41.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/Projection.lean,
theorems proj_proj, dot_sub_proj,
proj_sq_le, proj_add_orth,
proj_add, at observation-data-mining f3914f0; what the
check covers is stated in the book’s appendix
C.
Used in
Data Mining as Observation primer L, chapters 0, 1, 3, 4, 6, 7, 9, 11.
Related
dot product; read subspace; quotient; nuisance.
See also
Book equations stated beside the entry’s terms, not defining it: 0.12a, 0.12b.
Ledger rows that cite the entry’s records without naming it: GO-1.
Sources-table rows that share a record with the entry without naming it: chapter 6 section 6.2, chapter 11 section 11.7.
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.