Are homogenous systems of equations with a trivial solution always consistent?

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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 ?




linear-algebra




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  • 1




    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.
    – Sriram Gopalakrishnan
    Aug 1 at 2:53






  • 1




    Also your matrix of coefficients is row equivalent to identity matrix
    – Panchal Shamsundar
    Aug 1 at 2:59










  • 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 :)
    – Aditi
    Aug 1 at 3:02














up vote
0
down vote

favorite












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 ?




linear-algebra




share|cite|improve this question















  • 1




    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.
    – Sriram Gopalakrishnan
    Aug 1 at 2:53






  • 1




    Also your matrix of coefficients is row equivalent to identity matrix
    – Panchal Shamsundar
    Aug 1 at 2:59










  • 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 :)
    – Aditi
    Aug 1 at 3:02












up vote
0
down vote

favorite









up vote
0
down vote

favorite











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 ?




linear-algebra




share|cite|improve this question











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 ?





linear-algebra





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Aug 1 at 2:45









Aditi

677314




677314







  • 1




    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.
    – Sriram Gopalakrishnan
    Aug 1 at 2:53






  • 1




    Also your matrix of coefficients is row equivalent to identity matrix
    – Panchal Shamsundar
    Aug 1 at 2:59










  • 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 :)
    – Aditi
    Aug 1 at 3:02












  • 1




    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.
    – Sriram Gopalakrishnan
    Aug 1 at 2:53






  • 1




    Also your matrix of coefficients is row equivalent to identity matrix
    – Panchal Shamsundar
    Aug 1 at 2:59










  • 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 :)
    – Aditi
    Aug 1 at 3:02







1




1




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.
– Sriram Gopalakrishnan
Aug 1 at 2:53




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.
– Sriram Gopalakrishnan
Aug 1 at 2:53




1




1




Also your matrix of coefficients is row equivalent to identity matrix
– Panchal Shamsundar
Aug 1 at 2:59




Also your matrix of coefficients is row equivalent to identity matrix
– Panchal Shamsundar
Aug 1 at 2:59












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 :)
– Aditi
Aug 1 at 3:02




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 :)
– Aditi
Aug 1 at 3:02










1 Answer
1






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up vote
1
down vote



accepted










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.






share|cite|improve this answer





















  • Thank you for the answer !
    – Aditi
    Aug 1 at 2:53










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote



accepted










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.






share|cite|improve this answer





















  • Thank you for the answer !
    – Aditi
    Aug 1 at 2:53














up vote
1
down vote



accepted










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.






share|cite|improve this answer





















  • Thank you for the answer !
    – Aditi
    Aug 1 at 2:53












up vote
1
down vote



accepted







up vote
1
down vote



accepted






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.






share|cite|improve this answer













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.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Aug 1 at 2:51









Brian Fitzpatrick

20.7k42957




20.7k42957











  • Thank you for the answer !
    – Aditi
    Aug 1 at 2:53
















  • Thank you for the answer !
    – Aditi
    Aug 1 at 2:53















Thank you for the answer !
– Aditi
Aug 1 at 2:53




Thank you for the answer !
– Aditi
Aug 1 at 2:53












 

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