Problem 1
Solve c₁+c₂=5 and c₁−c₂=1; c₁=3,c₂=2, so coordinates are (3,2).
Faculty use
Coordinate Vectors Relative to a Basis
Reward correct mathematics, explicit dimensions/hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
The vector is unchanged; only its coordinate representation changes.
Problem 3
Definitions should identify the vector/matrix/space/transformation, dimensions or domain/codomain, and the hypotheses behind representing the same vector in different coordinate systems. Use the formulas v=c₁b₁+⋯+cₙbₙ; [v]_B=(c₁,…,cₙ).
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 feature selection, efficient representation, degrees of freedom, and coordinate 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.