4
$\begingroup$

If {$v_1,v_2,v_3$} are linear independent is {$v_1+3v_3,v_2+v_1,v_1+v_3$} also linear independent ?

I tried $ a(v_1+3v_3)+b(v_2+v_1) +c(v_1+v_3)=0$

thus by grouping i got $(a+b+c)v_1+(3a+c)v_3+bv_2=0$

thus because $v_1,v_3,v_2$ are linear independent this has only a solution if $a,b, c $ are 0

$\endgroup$
  • 1
    $\begingroup$ Yes. WLOG you can assume $v_i = e_i$ where $e_i$ is the canonical basis, after that you just have a determinant to compute. $\endgroup$ – user171326 Jan 13 '17 at 0:51
  • $\begingroup$ $V \supseteq S$ is lin ind if $\forall x \in S: x \notin <S-\{x\}>$.. $\endgroup$ – mle Feb 15 '17 at 22:29
4
$\begingroup$

Your solution is correct; however, I'd suggest inserting one piece of detail that you skipped over.

The fact that $$ (a+b+c)v_1+bv_2+(3a+c)v_3=0 $$ does not, immediately, imply that you only have a solution for $a=b=c=0$; it has a solution only for $a+b+c=0$, $b=0$, and $3a+c=0$.

Now, clearly, $b=0$; then the system reduces to $a+c=0$ and $3a+c=0$, and you can finish the algebra from here to show that $a=b=c=0$.

$\endgroup$

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.