1. In a linear system Ax = b where A is a row echelon, a pivot in a row of A is the first non-zero entry (if any) of that row; if a column of A does not contain a pivot, the corresponding unknown is called a free unknown. The goal of this question is to prove the following claim:

Proposition 1. Assume that a linear system Ax = b satisfies the following two conditions: (1) A is a row echelon. (2) If all entries in the jth row of A are zero, then the jth row of b is also zero. (This holds for all j.) Then for any values assigned to the free unknowns of the system, the system has a unique solution in which the values of the free unknowns are as assigned. (a) Does the proposition hold without Condition (2)? (b) Daniel suggests the following proof of the proposition