Linear Algebra: Case Studies

Duplicate sensor features

A model uses t and 2t as separate features. Two solvers return different coefficients with the same predictions. Establish the null direction, show that predictions are identifiable while individual slope coefficients are not, and document the minimum-norm or penalty convention. Do not call either coefficient set uniquely learned from the data.

Small residual, uncertain calibration

Two nearly parallel constraints are measured with four decimal places. Exact solves for two plausible right-hand sides differ substantially. Report both measurement sensitivity and arithmetic error. A stable solver cannot establish which measurement is correct; a rounded residual of zero is not a data-quality check.

A discarded low-energy direction

An SVD truncation removes a small singular value, reducing storage or noise amplification. Evaluate discarded energy and the downstream quantity separately. Use P17 to show that low output energy can correspond to a meaningful coefficient. Document the cutoff, units, alternative model and decision evidence.

Each case requires a reproducible small input, mathematical explanation, numerical evidence and one changed-assumption result. Use the project to combine them into a defensible report.