[Solved] 18.06 Exercise 10-The four fundamental subspaces

Problem 10.1: (3.6 #11. Introduction to Linear Algebra: Strang) A is an m by n matrix of rank r. Suppose there are right sides b for which Ax = b has no solution.

1. What are all the inequalities (< or ≤) that must be true between m, n, and r?
2. How do you know that A^Ty = 0 has solutions other than y = 0?

Problem 10.2: (3.6 #24.) A^Ty = d is solvable when d is in which of the four subspaces? The solution y is unique when the contains only the zero vector.