Basis, Rank, and Transformation Workshop

Complete P6–P10 independently. Derive coordinate formulas from maps before using them.

P6. Tall basis. Let B have columns (1,0,1) and (0,1,1). Find coordinates of (3,2,5); decide whether (0,0,1) has coordinates in that basis. Explain why B^-1 is not a legal expression for a square inverse.

P7. Same shear, new coordinates. T(x,y)=(x+y,y). Use the basis ((1,1),(1,−1)) on both sides. Derive P^-1 A P and check both transformed columns. Would the same formula work for unrelated input and output bases?

P8. Thin Q. For Q=[[1,0],[0,1],[0,0]], calculate Q^TQ and QQ^T. Interpret both products on vectors. Prove the latter is symmetric and idempotent.

P9. Projection optimality. Fit c+mt to (t,y)=(0,1),(1,2),(2,2). Give coefficients, fitted values, residual and squared error. Prove optimality using orthogonality and Pythagoras; a solver output alone is insufficient.

P10. Determinant and kernel. For A=[[1,2,3],[2,4,6],[0,1,5]], explain why the determinant is zero and verify (7,−5,1) lies in the nullspace. Test the incorrect candidate (2,−1,0). Does a tiny nonzero determinant imply sensitivity?

After your attempt, use the separate feedback.