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Phase 3 — Quizzes¶
🇪🇸 Espejo legible de
data/quizzes/phase-03-linear-algebra.yaml. Los ejemplos numéricos usan el grid de §A13 (verbos × formas) cuando aportan contexto.
Source of truth: data/quizzes/phase-03-linear-algebra.yaml.
q-03-01 — Why is SVD's rank the count of non-zero σ?¶
- Because U and V are orthogonal and orthogonal matrices have rank n, so rank(A) is determined by Σ alone; Σ has r non-zero pivots, hence rank r.
- Because the determinant is the product of singular values.
- Because the trace is the sum of singular values.
- Because rank is always equal to min(M, N) for any SVD.
Answer
**Choice 1.** Orthogonal multiplication preserves rank; rank(A) = rank(Σ) = count of non-zero diagonal entries.q-03-02 — Norms and their roles (multi-choice)¶
- L2 of a vector = sqrt of self inner product.
- Frobenius of a matrix = sqrt of sum of squared singular values.
- L1 of a vector encourages sparsity as a regularizer (corners at axes).
- Spectral norm of a matrix = largest singular value.
- Frobenius of a matrix = its largest singular value.
Answer
**Choices 1, 2, 3, 4.** Choice 5 confuses Frobenius (sum of σ²) with spectral (max σ).q-03-03 — Norms of v = [3, -4, 12, 0] (free)¶
Answer
L1 = 3 + 4 + 12 + 0 = **19**. L2 = sqrt(9 + 16 + 144 + 0) = sqrt(169) = **13**. L∞ = max(3, 4, 12, 0) = **12**.q-03-04 — Condition number and stability (free)¶
Answer
Solving `A x = b` amplifies relative perturbation by up to κ(A): `||δx||/||x|| ≤ κ · ||δb||/||b||`. A matrix with κ ≈ 10⁶ effectively loses 6 digits of precision in x compared to b.q-03-05 — Eckart-Young¶
- Singular values are diagonal entries of a covariance matrix.
- Each σ_i is the L2-length of the projection of A onto its i-th singular direction; bigger σ means more of A's "mass" lives there, so dropping a small σ loses little Frobenius energy.
- The largest σ is always 1 by construction.
- SVD is unique only when singular values are distinct.