Problem 1
{(1,2)}.
Faculty use
Row Space, Left Nullspace & Orthogonality
Reward correct mathematics, explicit dimensions/hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
y is orthogonal to every column of A and represents a relation among the rows/output constraints.
Problem 3
Definitions should identify the vector/matrix/space/transformation, dimensions or domain/codomain, and the hypotheses behind connecting row information to constraints and the transpose. Use the formulas Row(A)=Col(Aᵀ); N(Aᵀ)=Row(A)⊥.
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 solution structure, function approximation, matrix spaces, networks, and abstract modelling, 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.