0
$\begingroup$

If a homogeneous system of equations has only a trivial solution , can we call it consistent ?

For example , consider

$a_1x+b_1y+c_1z=0$

$a_2x+ b_2y +c_2z=0$

$a_3x+b_3y+c_3z=0$

Regardless of the values of the coefficents, $(0,0,0)$ will always be a solution of the above system of equations. Now we make an assumption that the system has only a trivial solution. Would we call these equations consistent in that case ?

$\endgroup$
  • 1
    $\begingroup$ Consistency is nothing more than the existence of at least one solution. The trivial solution is a solution, so any homogeneous system is necessarily consistent. $\endgroup$ – Sriram Gopalakrishnan Aug 1 '18 at 2:53
  • 1
    $\begingroup$ Also your matrix of coefficients is row equivalent to identity matrix $\endgroup$ – Tutankhamun Aug 1 '18 at 2:59
  • $\begingroup$ Thank you ! But I was confused whether (0,0,0) would classify for consistently since we call it trivial. But now I know the answer :) $\endgroup$ – Aditi Aug 1 '18 at 3:02
1
$\begingroup$

The term consistent is used to describe a system that has at least one solution. As you mention, every homogeneous system is solved by the trivial solution. This means that every homogeneous system is consistent.

$\endgroup$

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.