Problem 1
Not necessarily; an ill-conditioned A can convert a small residual into a large solution error.
Faculty use
Norms, Backward Error & Verification
Reward correct mathematics, explicit dimensions/hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Include method, norm, residual, scaling, pivoting, condition evidence, and a comparison or reconstruction check.
Problem 3
Definitions should identify the vector/matrix/space/transformation, dimensions or domain/codomain, and the hypotheses behind distinguishing a small residual from a small solution error. Use the formulas relative residual=‖b−Ax̂‖/‖b‖; κ(A)=‖A‖‖A⁻¹‖.
Problem 4
A strong error analysis identifies the first step that violates a definition, dimension, theorem hypothesis, or matrix-operation rule; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible matrix/vector model connected to scientific computing, repeated simulation, numerical software, and large structured systems, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: reconstruction, residual, RREF comparison, determinant/rank test, orthogonality, substitution, eigenpair check Av=λv, factorization product, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.