Find vectors to satisfy given dimensions of subspace spanned by 24 vectors that are rearrangements of 4-tuples. I have a homework exercise that I am having a difficult time understanding/visualizing:
Choose $x = (x_1, x_2, x_3, x_4)$ in $\mathbb{R}^4$. It has $24$ rearrangements like $(x_2, x_1, x_3, x_4)$ and $(x_4, x_3, x_1, x_2)$. Those $24$ vectors including $x$ itself span a subspace $S$. Find specific vectors $x$ so that dimension of $S$ is $0$, $1$, $3$, $4$.
I appreciate any help that can be given. Thanks! Source: 3.5.42, P183, Intro to Lin Alg, by G Strang 

$0$ is easy, not much choice. 
As to $1$, think of a nonzero vector that doesn't change when rearranged. 
As to $4$, think of the standard basis. 
As to $3$, think of the subspace $x_1+x_2+x_3+x_4= 0$. 
