Dimension problem (linear algebra)

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?

Quote:

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Originally Posted by faure72

The rank is 5, so does this mean that the dimension of S is also 5?

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yes