subj.

Vectors x1,...   xn are linery dependent iff
there exist numbers  a1,...an such that a1^2 + ... + an^2 !
= 0
and a1*x1+a2*x2 .... + an*xn = 0

Vectors are independent otherwise.

as your explanation 0 0 3 is a linery independed vector,isn't it?

Can I consider it like this:
Vectors x1,... xn are linery dependent iff

for any two vectors (xi,xj) in {x1,x2...xn}, there doesn't exist a real number K that
x[i,l]*K=x[j,l] (l=1,2...n).

Please help me.
Thanks very much!

x1=(1;0;0)
x2=(0;1;1)
x3=(1;1;1)

Vectors v1,v2,...,vn are linearly independent iff for any vi, you CAN'T find a set of real numbers (a1,a2,...,a(i-1),a(i+1),...,an) such that vi=a1*v1+a2*v2+...+a(i-1)*v(i-1)+a(i+1)*v(i+1)+...+an*vn.
That means for any n-dimensional vector v, you CAN find a set of real numbers (a1,a2...an) such that v=a1*v1+a2*v2+...+an*vn.