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Independent Linear Algebra Checkpoint

Work without notes or reference code. Show reasoning and assumptions. Calculators may check arithmetic after the model and proof are written. Give exact values where possible; state norms and tolerances for numerical claims.

  1. C1: Solve A=[[0,1],[2,2]], b=(3,8), showing the pivot exchange. Then certify inconsistency of [[1,1],[2,2]]x=(1,3) without trusting a solver.
  2. C2: For A=[[1,2,0],[0,0,1]], give bases and dimensions for column space and nullspace. Prove the nullspace description. Why must a subspace test include nonemptiness or the zero vector?
  3. C3: A=diag(2,1), B=[[1,1],[0,1]]. Derive its representation in basis B and apply it to coordinates (1,2). Verify in standard coordinates. Explain why adding a nonzero translation changes linearity.
  4. C4: Project b=(2,0,1) onto a=(1,1,0). Give the residual and a proof of optimality. If the design columns become a and 2a, describe all coefficient minimizers and identify the one of minimum Euclidean norm.
  5. C5: Compare determinant and 2-norm conditioning for 10^-6*I₂ and diag(10^6,10^-6). Explain the consequence of a negative determinant separately.
  6. C6: For diag(−4,2), explain what power iteration starting at (0,1) certifies. Give a spectral-radius-one example with unbounded powers and explain why a real rotation need not have a real diagonalization.
  7. C7: Prove [[3,1],[1,3]] is PD and give its Cholesky factor. Give a symmetric positive-diagonal matrix that is not PD and a counterexample to the nonnegative-leading-minors test for PSD.
  8. C8: For diag(5,2,0), give rank, minimum-norm pseudoinverse, rank-one approximation and both error norms. Explain full, reduced and compact shapes; state how a numerical cutoff could change the pseudoinverse.
  9. C9: A=diag(1,1/1000), b=(1,1/1000), proposed x=(1,0). Calculate residual, relative residual, normwise backward error, infinity-norm condition number and the forward-error bound. Compare with the actual relative error.
  10. C10: Fit a constant to (1,4) with weights (1,2), then separately fit (2,4) unweighted with ridge penalty 2*c². Compare ordinary least squares. Explain why solver accuracy cannot justify an unreported objective change, and when QR, Cholesky or an SVD-based method fits the stated problem.

After the attempt use scoring and derivations, then remediation. Finish the project on separate evidence.