For row reduced echelon matrix for a homogeneous system of equations, what solution would be there for r=n...
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I was learning Linear Algebra from Hoffman and Kunze. There the authors prove that for a row reduced echelon matrix with rows r and columns n for a homogeneous system of equations X, if r < n, X has a non-trivial solution, which I understood (substituting, r unknowns with remaining unknowns). However, what would happen in case of
- r = n
- r > n
For r = n, I am guessing it would be trivial solution only as no substitution is possible. Please correct me if wrong.
linear-algebra matrices homogeneous-equation
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add a comment |
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I was learning Linear Algebra from Hoffman and Kunze. There the authors prove that for a row reduced echelon matrix with rows r and columns n for a homogeneous system of equations X, if r < n, X has a non-trivial solution, which I understood (substituting, r unknowns with remaining unknowns). However, what would happen in case of
- r = n
- r > n
For r = n, I am guessing it would be trivial solution only as no substitution is possible. Please correct me if wrong.
linear-algebra matrices homogeneous-equation
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Well, just think how many unknowns are remaining to "substitute"?
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– Aniruddha Deshmukh
Dec 28 '18 at 6:10
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What is rank of the matrix?Is it r or less than than r?
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– ASHWINI SANKHE
Dec 28 '18 at 6:59
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@ASHWINISANKHE It's r. I can't seem to correct my question. It should be like : reduced echelon matrix with with non-zero rows r. Sorry for the omission.
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– dheeraj suthar
Dec 28 '18 at 7:28
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@AniruddhaDeshmukh I believe it should be zero for both case 1. and 2. So I think trivial solutions in case 2 also.
$endgroup$
– dheeraj suthar
Dec 28 '18 at 7:54
add a comment |
$begingroup$
I was learning Linear Algebra from Hoffman and Kunze. There the authors prove that for a row reduced echelon matrix with rows r and columns n for a homogeneous system of equations X, if r < n, X has a non-trivial solution, which I understood (substituting, r unknowns with remaining unknowns). However, what would happen in case of
- r = n
- r > n
For r = n, I am guessing it would be trivial solution only as no substitution is possible. Please correct me if wrong.
linear-algebra matrices homogeneous-equation
$endgroup$
I was learning Linear Algebra from Hoffman and Kunze. There the authors prove that for a row reduced echelon matrix with rows r and columns n for a homogeneous system of equations X, if r < n, X has a non-trivial solution, which I understood (substituting, r unknowns with remaining unknowns). However, what would happen in case of
- r = n
- r > n
For r = n, I am guessing it would be trivial solution only as no substitution is possible. Please correct me if wrong.
linear-algebra matrices homogeneous-equation
linear-algebra matrices homogeneous-equation
asked Dec 28 '18 at 6:06
dheeraj suthardheeraj suthar
72
72
$begingroup$
Well, just think how many unknowns are remaining to "substitute"?
$endgroup$
– Aniruddha Deshmukh
Dec 28 '18 at 6:10
$begingroup$
What is rank of the matrix?Is it r or less than than r?
$endgroup$
– ASHWINI SANKHE
Dec 28 '18 at 6:59
$begingroup$
@ASHWINISANKHE It's r. I can't seem to correct my question. It should be like : reduced echelon matrix with with non-zero rows r. Sorry for the omission.
$endgroup$
– dheeraj suthar
Dec 28 '18 at 7:28
$begingroup$
@AniruddhaDeshmukh I believe it should be zero for both case 1. and 2. So I think trivial solutions in case 2 also.
$endgroup$
– dheeraj suthar
Dec 28 '18 at 7:54
add a comment |
$begingroup$
Well, just think how many unknowns are remaining to "substitute"?
$endgroup$
– Aniruddha Deshmukh
Dec 28 '18 at 6:10
$begingroup$
What is rank of the matrix?Is it r or less than than r?
$endgroup$
– ASHWINI SANKHE
Dec 28 '18 at 6:59
$begingroup$
@ASHWINISANKHE It's r. I can't seem to correct my question. It should be like : reduced echelon matrix with with non-zero rows r. Sorry for the omission.
$endgroup$
– dheeraj suthar
Dec 28 '18 at 7:28
$begingroup$
@AniruddhaDeshmukh I believe it should be zero for both case 1. and 2. So I think trivial solutions in case 2 also.
$endgroup$
– dheeraj suthar
Dec 28 '18 at 7:54
$begingroup$
Well, just think how many unknowns are remaining to "substitute"?
$endgroup$
– Aniruddha Deshmukh
Dec 28 '18 at 6:10
$begingroup$
Well, just think how many unknowns are remaining to "substitute"?
$endgroup$
– Aniruddha Deshmukh
Dec 28 '18 at 6:10
$begingroup$
What is rank of the matrix?Is it r or less than than r?
$endgroup$
– ASHWINI SANKHE
Dec 28 '18 at 6:59
$begingroup$
What is rank of the matrix?Is it r or less than than r?
$endgroup$
– ASHWINI SANKHE
Dec 28 '18 at 6:59
$begingroup$
@ASHWINISANKHE It's r. I can't seem to correct my question. It should be like : reduced echelon matrix with with non-zero rows r. Sorry for the omission.
$endgroup$
– dheeraj suthar
Dec 28 '18 at 7:28
$begingroup$
@ASHWINISANKHE It's r. I can't seem to correct my question. It should be like : reduced echelon matrix with with non-zero rows r. Sorry for the omission.
$endgroup$
– dheeraj suthar
Dec 28 '18 at 7:28
$begingroup$
@AniruddhaDeshmukh I believe it should be zero for both case 1. and 2. So I think trivial solutions in case 2 also.
$endgroup$
– dheeraj suthar
Dec 28 '18 at 7:54
$begingroup$
@AniruddhaDeshmukh I believe it should be zero for both case 1. and 2. So I think trivial solutions in case 2 also.
$endgroup$
– dheeraj suthar
Dec 28 '18 at 7:54
add a comment |
1 Answer
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$Ax=0$ is homogeneous system.It always have trivial solution.If r=n then system will have trivial solution.As $rank(A)le min{(r,n)}$. Hence r can not be greater than n.
$endgroup$
$begingroup$
Thanks. I have not yet formally gone through the rank concept. But a quick search showed your assumption is true ( rank(A)≤min(r,n) ) . So both case 1 and 2 would have only trivial solutions. I guess I would have much better understanding once I go through the proof for rank(A)≤min(r,n)
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– dheeraj suthar
Dec 28 '18 at 8:55
add a comment |
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1 Answer
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1 Answer
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$begingroup$
$Ax=0$ is homogeneous system.It always have trivial solution.If r=n then system will have trivial solution.As $rank(A)le min{(r,n)}$. Hence r can not be greater than n.
$endgroup$
$begingroup$
Thanks. I have not yet formally gone through the rank concept. But a quick search showed your assumption is true ( rank(A)≤min(r,n) ) . So both case 1 and 2 would have only trivial solutions. I guess I would have much better understanding once I go through the proof for rank(A)≤min(r,n)
$endgroup$
– dheeraj suthar
Dec 28 '18 at 8:55
add a comment |
$begingroup$
$Ax=0$ is homogeneous system.It always have trivial solution.If r=n then system will have trivial solution.As $rank(A)le min{(r,n)}$. Hence r can not be greater than n.
$endgroup$
$begingroup$
Thanks. I have not yet formally gone through the rank concept. But a quick search showed your assumption is true ( rank(A)≤min(r,n) ) . So both case 1 and 2 would have only trivial solutions. I guess I would have much better understanding once I go through the proof for rank(A)≤min(r,n)
$endgroup$
– dheeraj suthar
Dec 28 '18 at 8:55
add a comment |
$begingroup$
$Ax=0$ is homogeneous system.It always have trivial solution.If r=n then system will have trivial solution.As $rank(A)le min{(r,n)}$. Hence r can not be greater than n.
$endgroup$
$Ax=0$ is homogeneous system.It always have trivial solution.If r=n then system will have trivial solution.As $rank(A)le min{(r,n)}$. Hence r can not be greater than n.
answered Dec 28 '18 at 7:58
ASHWINI SANKHEASHWINI SANKHE
12710
12710
$begingroup$
Thanks. I have not yet formally gone through the rank concept. But a quick search showed your assumption is true ( rank(A)≤min(r,n) ) . So both case 1 and 2 would have only trivial solutions. I guess I would have much better understanding once I go through the proof for rank(A)≤min(r,n)
$endgroup$
– dheeraj suthar
Dec 28 '18 at 8:55
add a comment |
$begingroup$
Thanks. I have not yet formally gone through the rank concept. But a quick search showed your assumption is true ( rank(A)≤min(r,n) ) . So both case 1 and 2 would have only trivial solutions. I guess I would have much better understanding once I go through the proof for rank(A)≤min(r,n)
$endgroup$
– dheeraj suthar
Dec 28 '18 at 8:55
$begingroup$
Thanks. I have not yet formally gone through the rank concept. But a quick search showed your assumption is true ( rank(A)≤min(r,n) ) . So both case 1 and 2 would have only trivial solutions. I guess I would have much better understanding once I go through the proof for rank(A)≤min(r,n)
$endgroup$
– dheeraj suthar
Dec 28 '18 at 8:55
$begingroup$
Thanks. I have not yet formally gone through the rank concept. But a quick search showed your assumption is true ( rank(A)≤min(r,n) ) . So both case 1 and 2 would have only trivial solutions. I guess I would have much better understanding once I go through the proof for rank(A)≤min(r,n)
$endgroup$
– dheeraj suthar
Dec 28 '18 at 8:55
add a comment |
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$begingroup$
Well, just think how many unknowns are remaining to "substitute"?
$endgroup$
– Aniruddha Deshmukh
Dec 28 '18 at 6:10
$begingroup$
What is rank of the matrix?Is it r or less than than r?
$endgroup$
– ASHWINI SANKHE
Dec 28 '18 at 6:59
$begingroup$
@ASHWINISANKHE It's r. I can't seem to correct my question. It should be like : reduced echelon matrix with with non-zero rows r. Sorry for the omission.
$endgroup$
– dheeraj suthar
Dec 28 '18 at 7:28
$begingroup$
@AniruddhaDeshmukh I believe it should be zero for both case 1. and 2. So I think trivial solutions in case 2 also.
$endgroup$
– dheeraj suthar
Dec 28 '18 at 7:54