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.



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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.