Problem 1
Apply T to bⱼ and express the result in codomain basis C.
Faculty use
Matrix Representations Relative to Bases
Reward correct mathematics, explicit dimensions/hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
A matrix records T relative to chosen coordinate bases; the underlying mapping is basis-independent.
Problem 3
Definitions should identify the vector/matrix/space/transformation, dimensions or domain/codomain, and the hypotheses behind encoding a transformation when domain and codomain bases are not standard. Use the formulas [T(v)]_C=[T]_{C←B}[v]_B; columns are [T(bⱼ)]_C.
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 graphics, coordinate changes, differential operators, data pipelines, and reversible models, 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.