Prove that any collection of vectors that includes zero vector can’t be linearly independent.

Question

I am stuck at the first step which is the definition of linear independence, that is, vectors $v_1, v_2, ......, v_k$ are linearly independent if and only if there exist scalars $c_1,c_2,.....c_k$ such that $ c_1v_1 + c_2v_2+ .......+ c_kv_k= 0 $ implying $ c_1=c_2=c_3=....=c_k=0.$

Now how do I proceed to prove the question about zero vector? The only thing I can think of is that with a zero vector it becomes impossible to prove that scalar multiplied with it is zero. Please help.

Answer 1

Linearly independent means

$c_1v_1+c_2v_2+\cdots c_iv_i+\cdots+c_kv_k=0\implies c_1=c_2=\cdots=c_k=0$.

But if $v_i=0$, then we have $0v_1+0v_2+\cdots+5v_i+\cdots+0v_k=0$.