Dimension and Basis (Linear Algebra)

So I put the vectors inside the span into a matrix like so:

1 1 1 2 1
1 1 2 1 1
1 2 1 1 1
2 1 1 1 1
1 1 1 1 2

Then I row reduced it to get

1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1


The rank is 5, so does this mean the dimension of S is 5? And is the "Remember the 3=0" part just a piece of unneeded information in the problem?

If your row reduction was done right, then yes. That row reduction shows that you have linearly independent vectors. Since V is 5 dimentional and S has 5 independent vectors, S is a basis for F₃⁵. (edit: I see you've deleted your extra work).