I am stuck on a math question for my linear algebra midterm regarding REF.
The problem says: Consider a system of linear equations with augmented matrix $[ A \ \ |\ \ \vec b ]$. If the system has infinitely many solutions, then any row echelon form $[ A \ \ |\ \ \vec b ]$ must contain a row of zeros. True or False?
I said true since 0x = 0 would have infinite amount of solutions. However, according to the solution sheet, it says it's false.
The solution: "This statement if FALSE. The system with augmented matrix
1, 0, 2 | 1
0, 1, −3 | −1
is in REF and is consistent with infinitely many solutions (the third variable is free) but has no row of zeros."
I don't get how this has infinite amount of solutions. My understanding was that something like 0x=0 would have infinite amount of solutions since any value of x would make the the equation equal to 0.
Can someone explain? How does the free variable play into this? Thanks!