Show that any finite subset, T, of V which contains S is also linearly dependent.

Question

I'm currently stuck on this question.

Let V be a vector space, and let S be a finite set of vectors in V which is linearly dependent. Show that any finite subset, T, of V which contains S is also linearly dependent.

Linear algebra this year has been rather difficult for me, particularly the proofs. Any hints or help would be greatly appreciated.

Thanks.

Answer 1

Let$$ c_1s_1+...+c_ks_k=0 $$ , where not all $c_k$=0.

Extend to all $t_j:=t_1,..,t_m \in T-S$, with all 0 coefficients :

$ c_1s_1+...+c_ks_k+ 0t_1+0t_2+...+0t_m= (c_1s_1+...+c_ks_k)+ 0t_1+0t_2+...+0t_m=0=$

$0+0(t_1+t_2+..+t_m)=0+0=0$ Is a non-trivial linear combination in T that adds to $0$,

so that the set $T$ is linearly-dependent.