Problem 1
q₁=(1,0); subtract projection to get u₂=(0,1), so q₂=(0,1).
Faculty use
Gram–Schmidt Orthonormalization
Reward correct mathematics, explicit dimensions/hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Nearly dependent vectors can lose orthogonality through rounding; modified Gram–Schmidt or Householder QR is more stable.
Problem 3
Definitions should identify the vector/matrix/space/transformation, dimensions or domain/codomain, and the hypotheses behind converting an independent list into an orthonormal basis. Use the formulas u₂=v₂−proj_{u₁}v₂; qᵢ=uᵢ/‖uᵢ‖.
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 data fitting, approximation, signal decomposition, statistics, and numerical stability, 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.