There's a much simpler way to answer the question, that can be done with no computation. Answer the following questions and it will be obvious to you.
What's the dimension of R3 (i.e., dim(R3))?
How many vectors does it take to span R3?
How many vectors are there in S?
Does S span R3?
Going back to the work you did, you have a lot of extra stuff that doesn't make any sense to me. A given set of vectors spans R3 if any vector in R3 is some linear combination of the vectors in the set. IOW, for any vector <x, y, z>, there is a solution for the constants a and b in this equation:
a<1, 0, 1> + b<0, 2, 5> = <x, y, z>
Setting this up as an augmented matrix gives you this:
1 0 | x
0 2 | y
1 5 | z
After row reduction, I get j
1 0 | x
0 1 | y/2
0 0 | z-x -5y/2
The first two rows say that a = x and b = y/2, but the bottom row says that 0a + 0b = z - x - 5y/2. This last equation is saying that the system of equations has a solution only if z - x -5y/2 = 0. IOW, for some vectors <x, y, z> there is no solution.