Problem 1
Dual spaces, quotient spaces, invariant subspaces, Jordan/rational canonical form, multilinear algebra, or operators.
Faculty use
Second-Course Bridge: Abstract, Numerical & Applied Directions
Reward correct mathematics, explicit dimensions/hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Krylov methods, numerical optimization, tensor methods, machine learning, control, inverse problems, or PDE discretization.
Problem 3
Definitions should identify the vector/matrix/space/transformation, dimensions or domain/codomain, and the hypotheses behind mapping first-course mastery into advanced linear algebra, numerical analysis, data science, and differential equations. Use the formulas invariant subspaces and canonical forms; iterative eigensolvers and optimization.
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 large systems, reproducible computation, research communication, ethical data modelling, and advanced mathematics, 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.