Elimination and Subspace Diagnostics Lab

Complete P1–P5 without the reference implementation. Keep the first attempt and state dimensions, field and proof obligations. Use exact fractions.

P1. Pivot exchange. Solve A=[[0,1],[1,0]], b=(2,3). Explain the first row exchange, determinant sign and why a zero first entry does not imply singularity.

P2. Consistency certificate. For A=[[1,2,3],[2,4,6]], first take b=(1,2), then b=(1,3). Give rank, a particular solution and nullspace basis in the first case. Give y with y^T A=0, y^T b!=0 in the second. Prove the certificate's meaning.

P3. Subspace boundary. Classify x+y+z=0, x+y+z=1, the empty set and the union of two distinct coordinate axes. Supply closure proofs or counterexamples. Explain why a universal closure statement alone can accidentally admit the empty set.

P4. Four spaces. For A=[[1,0,1],[0,1,1]], describe all four fundamental subspaces, their dimensions and ambient spaces. Prove one orthogonal-complement relation.

P5. Original columns. Reduce A=[[1,2],[2,4]]. Compare the original pivot column with the reduced pivot column; show they span different subsets of R². Explain exactly what row operations preserve.

After your attempt, use the separate feedback.