Problem 1
Show U⊆W and W⊆U.
Faculty use
Proof Strategies in Linear Algebra
Reward correct mathematics, explicit dimensions/hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Use nonzero matrices with incompatible image/nullspace, such as A=[[1,0],[0,0]], B=[[0,0],[0,1]], for which AB=0.
Problem 3
Definitions should identify the vector/matrix/space/transformation, dimensions or domain/codomain, and the hypotheses behind choosing direct proof, contrapositive, contradiction, dimension, or counterexample. Use the formulas prove both inclusions for subspace equality; disprove universal claim with one counterexample.
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.