Bibliographic record
Abstract
Let V be a vector space.Let B = {t»i, . . ., b TO } and E = { e\, . . ., e n } be two bases for V.This means that both B and E are linearly independent and V = Span B = Span E. Our objective is to prove that m = n.This fundamental result tells us that the number of elements in any basis for a finite-dimensional vector space is the same.This number is thus well defined and is called the dimension of the vector space.One way to prove this result is to use a well-known result about linear systems of equations.Consider a homogeneous linear system of equations.If the number of unknowns is more than the number of equations, then this homogeneous system has nonzero solutions.This theorem implies that in R™, a set that contains more than n vectors cannot be linearly independent.Then the result about dimension follows easily.Instead of appealing to this argument, we shall provide a self-contained vectorial proof.The arguments used in this proof are purely algebraic.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.175 | 0.047 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".