For row reduced echelon matrix for a homogeneous system of equations, what solution would be there for r=n...
$begingroup$
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
asked Dec 28 '18 at 6:06
dheeraj suthardheeraj suthar
72
$endgroup$
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
asked Dec 28 '18 at 6:06
dheeraj suthardheeraj suthar
72
$endgroup$
$begingroup$
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 |
$begingroup$
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
asked Dec 28 '18 at 6:06
dheeraj suthardheeraj suthar
72
$endgroup$
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
asked Dec 28 '18 at 6:06
dheeraj suthardheeraj suthar
72
asked Dec 28 '18 at 6:06
dheeraj suthardheeraj suthar
72
asked Dec 28 '18 at 6:06
dheeraj suthardheeraj suthar
72
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
1
active
oldest
votes
$begingroup$
$Ax=0$ 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.
answered Dec 28 '18 at 7:58
ASHWINI SANKHEASHWINI SANKHE
11210
$endgroup$
$begingroup$
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 |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3054621%2ffor-row-reduced-echelon-matrix-for-a-homogeneous-system-of-equations-what-solut%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$Ax=0$ 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.
answered Dec 28 '18 at 7:58
ASHWINI SANKHEASHWINI SANKHE
11210
$endgroup$
$begingroup$
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.
answered Dec 28 '18 at 7:58
ASHWINI SANKHEASHWINI SANKHE
11210
$endgroup$
$begingroup$
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.
answered Dec 28 '18 at 7:58
ASHWINI SANKHEASHWINI SANKHE
11210
$endgroup$
$Ax=0$ 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
11210
answered Dec 28 '18 at 7:58
ASHWINI SANKHEASHWINI SANKHE
11210
answered Dec 28 '18 at 7:58
ASHWINI SANKHEASHWINI SANKHE
11210
answered Dec 28 '18 at 7:58
ASHWINI SANKHEASHWINI SANKHE
11210
11210
$begingroup$
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 |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3054621%2ffor-row-reduced-echelon-matrix-for-a-homogeneous-system-of-equations-what-solut%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
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