I did out the following problem below, but I'm not sure if it's correct so any help would be appreciated!

F_3 is the finite field with three elements, where the three elements are {0, 1, 2}. (Addition in F_3 is defined modulo 3, which is another way of saying that 3=0)

V = (F_3)^5 is a vector space.

Let S = Span({(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)} ⊂ V. What is the dimension of S? (Remember that 3=0)

I tried this and here's what I came up with:

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 that the dimension of S is also 5?