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?