Good night, i'm working in a problem, i need an basis and the dimension of the space.

$a\_{1}=(1,0,0,-1),\:a\_{2}=(2,1,1,0),\:a\_{3}=(1,1,1,1),\:a\_{4}=(1,2,3,4),\:a\_{5}=(0,1,2,3)$

I make this:

$\left[ \begin {array}{cccc} 1&0&0&-1\\ 2&1&1&0
\\ 1&1&1&1\\ 1&2&3&4
\\ 0&1&2&3\end {array} \right]$ 

and i apply gauss for reduce the matrix:

$ \left[ \begin {array}{cccc} 1&0&0&-1\\ 0&1&0&1
\\ 0&0&1&1\\ 0&0&0&0
\\ 0&0&0&0\end {array} \right]$ 

now, i have 3 linearly independent vectors and my dimensions is 3. but, i can take any vector and they go linearly independent? for example:
$a\_{1}=(1,0,0,-1),\:a\_{2}=(2,1,1,0),\:a\_{3}=(1,1,1,1)$

or

$a\_{3}=(1,1,1,1),\:a\_{4}=(1,2,3,4),\:a\_{5}=(0,1,2,3)$

and going to be linearly independent? thanks!!