Can a set containing a single vector from a vector space over a finite field be linearly dependent?

Question

Take the set $S=\{v=(1,1)\}\subset F_2 ^2$. $v+v=(0,0)$ is a linear combination of vectors from $S$. Is $S$ linearly dependent?

Answer 1

No, $S$ is linearly independent. We, indeed, have $v+v=(0,0)$, but given that here $-v=v$ that does not say any more about linear (in)dependence than the equation $v-v=(0,0)$ would in a vector space over reals.

Yet another way of saying the same thing is to observe that $$ v+v=1\cdot v+1\cdot v=(1+1)\cdot v =0\cdot v, $$ so the l.h.s. of $v+v=(0,0)$ is the trivial linear combination, where all the scalars are equal to zero.

Answer 2

An easier way to see the answer is: Suppose r*(1,1) = (0,0). Since r is a scalar in F(2), it can only be 0 or 1. If r = 0, then S is linearly independent. If r = 1, then 1*(1,1) = (1,1) not equal to (0,0). So S is independent.