How would you show that the vectors (1,1,0) (1,0,1) and (0,1,1) generate F^3?
Do you simply put them in the matrix and solve the matrix or?
Since F^3 (whatever that is) has dimension three it means if you can show {(1,1,0),(1,0,1),(0,1,1)} are linearly independent it would mean they must spam the space.
Now say $\displaystyle a(1,1,0)+b(1,0,1)+c(0,1,1) = (0,0,0)$.
This means,
$\displaystyle a+b = 0$
$\displaystyle a+c=0$
$\displaystyle b+c=0$
Now argue that $\displaystyle a=b=c=0$.