Did my textbook make a mistake, or am I just misunderstanding this concept (linear algebra)?

This is not homework; I am studying for an exam.

My textbook has a practice problem: “determine if b is a linear combination of a1, a2, and a3.”

Please note that all of the following matrices are supposed to be vertical, with 1 column and 3 rows, but of course the limitations of Fluther are such that I can’t write them that way.

a1 = [1 -2 0], a2 = [0 1 2], a3 = [5 -6 8], b = [2 -1 6]

I wrote the augmented matrix with a1, a2, a3, b as the columns and put it in reduced row echelon form. Here is my result (this cannot be where my mistake is because I had my calculator do it for me):

1 0 5 2
0 1 4 3
0 0 0 0

It is my understanding that because a solution exists (in fact, many solutions because x3 is free), b is a linear combination of a1, a2, a3. But my textbooks disagrees! Am I thinking about this wrong, or is my textbook wrong?