Condition for a Linear Equation System to have non-trivial Solution
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I have this Theorem in my book:
For a Homogeneous System of $m$ Linear Equations in $n$ unknowns, if $m lt n$, then the system has a non-trivial solution.
I have a confusion about the condition mentioned: Wouldn't it be $n lt m$ the condition for non-trivial solution? It seems to me that $m lt n$ is precisely the case we have either only trivial solution or no solution at all.
linear-algebra systems-of-equations
asked Jan 15 at 20:29
freehumoristfreehumorist
351214
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add a comment |
$begingroup$
I have this Theorem in my book:
For a Homogeneous System of $m$ Linear Equations in $n$ unknowns, if $m lt n$, then the system has a non-trivial solution.
I have a confusion about the condition mentioned: Wouldn't it be $n lt m$ the condition for non-trivial solution? It seems to me that $m lt n$ is precisely the case we have either only trivial solution or no solution at all.
linear-algebra systems-of-equations
asked Jan 15 at 20:29
freehumoristfreehumorist
351214
$endgroup$
add a comment |
$begingroup$
I have this Theorem in my book:
For a Homogeneous System of $m$ Linear Equations in $n$ unknowns, if $m lt n$, then the system has a non-trivial solution.
I have a confusion about the condition mentioned: Wouldn't it be $n lt m$ the condition for non-trivial solution? It seems to me that $m lt n$ is precisely the case we have either only trivial solution or no solution at all.
linear-algebra systems-of-equations
asked Jan 15 at 20:29
freehumoristfreehumorist
351214
$endgroup$
I have this Theorem in my book:
For a Homogeneous System of $m$ Linear Equations in $n$ unknowns, if $m lt n$, then the system has a non-trivial solution.
I have a confusion about the condition mentioned: Wouldn't it be $n lt m$ the condition for non-trivial solution? It seems to me that $m lt n$ is precisely the case we have either only trivial solution or no solution at all.
linear-algebra systems-of-equations
linear-algebra systems-of-equations
asked Jan 15 at 20:29
freehumoristfreehumorist
351214
asked Jan 15 at 20:29
freehumoristfreehumorist
351214
edited Jan 15 at 20:58
freehumorist
asked Jan 15 at 20:29
freehumoristfreehumorist
351214
asked Jan 15 at 20:29
freehumoristfreehumorist
351214
asked Jan 15 at 20:29
freehumoristfreehumorist
351214
351214
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Hint:
look at.
$2x=0$ with $m=1 , n=1$
and
$2x+y=0$ with $m=1, n=2$
what is the equation with non trivial ( i.e. not null) solution?
answered Jan 15 at 20:35
Emilio NovatiEmilio Novati
52.2k43474
$endgroup$
$begingroup$
Thank you, I can see now.
$endgroup$
– freehumorist
Jan 15 at 20:58
add a comment |
$begingroup$
The book's claim is wrong:
$$begin{cases}x-y=0,\3x+y=0,\x+y=0end{cases}$$ only has a trivial solution.
answered Jan 15 at 20:56
Yves DaoustYves Daoust
131k676229
$endgroup$
$begingroup$
Sorry for that. That s on me. I falsely copied the hypothesis. Edited.
$endgroup$
– freehumorist
Jan 15 at 20:58
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Hint:
look at.
$2x=0$ with $m=1 , n=1$
and
$2x+y=0$ with $m=1, n=2$
what is the equation with non trivial ( i.e. not null) solution?
answered Jan 15 at 20:35
Emilio NovatiEmilio Novati
52.2k43474
$endgroup$
$begingroup$
Thank you, I can see now.
$endgroup$
– freehumorist
Jan 15 at 20:58
add a comment |
$begingroup$
Hint:
look at.
$2x=0$ with $m=1 , n=1$
and
$2x+y=0$ with $m=1, n=2$
what is the equation with non trivial ( i.e. not null) solution?
answered Jan 15 at 20:35
Emilio NovatiEmilio Novati
52.2k43474
$endgroup$
$begingroup$
Thank you, I can see now.
$endgroup$
– freehumorist
Jan 15 at 20:58
add a comment |
$begingroup$
Hint:
look at.
$2x=0$ with $m=1 , n=1$
and
$2x+y=0$ with $m=1, n=2$
what is the equation with non trivial ( i.e. not null) solution?
answered Jan 15 at 20:35
Emilio NovatiEmilio Novati
52.2k43474
$endgroup$
Hint:
look at.
$2x=0$ with $m=1 , n=1$
and
$2x+y=0$ with $m=1, n=2$
what is the equation with non trivial ( i.e. not null) solution?
answered Jan 15 at 20:35
Emilio NovatiEmilio Novati
52.2k43474
answered Jan 15 at 20:35
Emilio NovatiEmilio Novati
52.2k43474
answered Jan 15 at 20:35
Emilio NovatiEmilio Novati
52.2k43474
answered Jan 15 at 20:35
Emilio NovatiEmilio Novati
52.2k43474
52.2k43474
$begingroup$
Thank you, I can see now.
$endgroup$
– freehumorist
Jan 15 at 20:58
add a comment |
$begingroup$
Thank you, I can see now.
$endgroup$
– freehumorist
Jan 15 at 20:58
$begingroup$
Thank you, I can see now.
$endgroup$
– freehumorist
Jan 15 at 20:58
$begingroup$
Thank you, I can see now.
$endgroup$
– freehumorist
Jan 15 at 20:58
add a comment |
$begingroup$
The book's claim is wrong:
$$begin{cases}x-y=0,\3x+y=0,\x+y=0end{cases}$$ only has a trivial solution.
answered Jan 15 at 20:56
Yves DaoustYves Daoust
131k676229
$endgroup$
$begingroup$
Sorry for that. That s on me. I falsely copied the hypothesis. Edited.
$endgroup$
– freehumorist
Jan 15 at 20:58
add a comment |
$begingroup$
The book's claim is wrong:
$$begin{cases}x-y=0,\3x+y=0,\x+y=0end{cases}$$ only has a trivial solution.
answered Jan 15 at 20:56
Yves DaoustYves Daoust
131k676229
$endgroup$
$begingroup$
Sorry for that. That s on me. I falsely copied the hypothesis. Edited.
$endgroup$
– freehumorist
Jan 15 at 20:58
add a comment |
$begingroup$
The book's claim is wrong:
$$begin{cases}x-y=0,\3x+y=0,\x+y=0end{cases}$$ only has a trivial solution.
answered Jan 15 at 20:56
Yves DaoustYves Daoust
131k676229
$endgroup$
The book's claim is wrong:
$$begin{cases}x-y=0,\3x+y=0,\x+y=0end{cases}$$ only has a trivial solution.
answered Jan 15 at 20:56
Yves DaoustYves Daoust
131k676229
answered Jan 15 at 20:56
Yves DaoustYves Daoust
131k676229
answered Jan 15 at 20:56
Yves DaoustYves Daoust
131k676229
answered Jan 15 at 20:56
Yves DaoustYves Daoust
131k676229
131k676229
$begingroup$
Sorry for that. That s on me. I falsely copied the hypothesis. Edited.
$endgroup$
– freehumorist
Jan 15 at 20:58
add a comment |
$begingroup$
Sorry for that. That s on me. I falsely copied the hypothesis. Edited.
$endgroup$
– freehumorist
Jan 15 at 20:58
$begingroup$
Sorry for that. That s on me. I falsely copied the hypothesis. Edited.
$endgroup$
– freehumorist
Jan 15 at 20:58
$begingroup$
Sorry for that. That s on me. I falsely copied the hypothesis. Edited.
$endgroup$
– freehumorist
Jan 15 at 20:58
add a comment |
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