All vectors are 4 dimension vectors, they can only span a 4 dimension space, the maximum rank is 4. There are 5 vectors, so, they must be linearly dependent.
Let A = [ a1; a2; a3; a4; a5] , which is:
1 -1 2 3
-2 4 -1 3
-1 2 0 3
0 6 2 3
2 -6 3 4
use matlab:
rank(A)
ans = 4
we know the rank is 4.
Moreover, you can verify that the rank of [a1;a2;a3;a4] is 4, too. They are linearly independent.