Linear Algebra: Mistake Clinic
Module-Specific Mistake Radar
| Mistake | Counterexample or repair |
|---|---|
| Count printed equations as rank | Duplicate one equation and compare pivots |
| Use reduced columns as original column basis | P5: (1,2) versus (1,0) |
| Invert a tall basis matrix | Solve Bc=v and check membership |
| Confuse a fitted vector with unique coefficients | Duplicate a design column |
| Infer numerical safety from determinant | Compare epsilon*I with diag(M,1/M) |
| Assume every power run finds the dominant mode | Start diag(3,1) at (0,1) |
| Infer PSD from nonnegative leading minors | diag(0,−1) |
| Treat a Frobenius tail sum as the norm | Take the square root of squared tail energy |
| Accept a raw small residual as accurate x | A=diag(1,0.001), P16/A9 |
| Treat regularization as an unchanged problem | Write the added penalty and its optimality condition |
Practice Mistake Checks
Pick three errors, construct a fresh small fixture for each, and state a corrected claim with its hypotheses. Save the first incorrect explanation and the repaired argument. Revisit practice feedback only after attempting the changed fixture; copying a previous numerical answer is not remediation.