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

For example , consider


$a_2x+ b_2y +c_2z=0$


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 ?

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    $\begingroup$ 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
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    $\begingroup$ 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

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.


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