Problem 1
x+2y−z=0; let y=s,z=t, so x=(−2s+t,s,t)=s(−2,1,0)+t(1,0,1).
Faculty use
Nullspace and Homogeneous Solutions
Reward correct mathematics, explicit dimensions/hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
If Ax=b and Az=0, then A(x+tz)=b for every t.
Problem 3
Definitions should identify the vector/matrix/space/transformation, dimensions or domain/codomain, and the hypotheses behind describing inputs sent to zero and directions of nonuniqueness. Use the formulas N(A)={x:Ax=0}; general solution=xₚ+N(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.