homogeneous only trivial or infinite
up vote
-2
down vote
favorite
This question has three parts. There are similar questions on stack exchange, but if you read all of the questions, then you'll see that this is not a duplicate [at least not one that I could find.]
From the Oregon math website:
"A $ntimes n$ homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions."
Does this imply that a homogeneous system $Ax=0$ has only a trivial solution or infinite number of solutions?
If not, then what values of the determinant for $A$ imply that there is a non-trivial, unique solution, for a homogeneous equation?
If it does imply that $Ax=0$ only has a trivial or infinite (free parameter) solution, isn't that kind of weird?
linear-algebra systems-of-equations determinant
add a comment |
up vote
-2
down vote
favorite
This question has three parts. There are similar questions on stack exchange, but if you read all of the questions, then you'll see that this is not a duplicate [at least not one that I could find.]
From the Oregon math website:
"A $ntimes n$ homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions."
Does this imply that a homogeneous system $Ax=0$ has only a trivial solution or infinite number of solutions?
If not, then what values of the determinant for $A$ imply that there is a non-trivial, unique solution, for a homogeneous equation?
If it does imply that $Ax=0$ only has a trivial or infinite (free parameter) solution, isn't that kind of weird?
linear-algebra systems-of-equations determinant
add a comment |
up vote
-2
down vote
favorite
up vote
-2
down vote
favorite
This question has three parts. There are similar questions on stack exchange, but if you read all of the questions, then you'll see that this is not a duplicate [at least not one that I could find.]
From the Oregon math website:
"A $ntimes n$ homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions."
Does this imply that a homogeneous system $Ax=0$ has only a trivial solution or infinite number of solutions?
If not, then what values of the determinant for $A$ imply that there is a non-trivial, unique solution, for a homogeneous equation?
If it does imply that $Ax=0$ only has a trivial or infinite (free parameter) solution, isn't that kind of weird?
linear-algebra systems-of-equations determinant
This question has three parts. There are similar questions on stack exchange, but if you read all of the questions, then you'll see that this is not a duplicate [at least not one that I could find.]
From the Oregon math website:
"A $ntimes n$ homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions."
Does this imply that a homogeneous system $Ax=0$ has only a trivial solution or infinite number of solutions?
If not, then what values of the determinant for $A$ imply that there is a non-trivial, unique solution, for a homogeneous equation?
If it does imply that $Ax=0$ only has a trivial or infinite (free parameter) solution, isn't that kind of weird?
linear-algebra systems-of-equations determinant
linear-algebra systems-of-equations determinant
edited Nov 22 at 18:22
Martin Sleziak
44.6k7115269
44.6k7115269
asked Nov 22 at 18:16
Bertram
41
41
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
1
down vote
It is good to notice that a homogeneous system $Ax=0$ always has at least zero solution $x=0$. (So for homogeneous system you cannot have no solutions at all.)
If you have any solution $x$, than any scalar multiple $cx$ is also a solution of the same homogeneous system. So once you have at least one non-zero solution, you immediately get many other solutions. (If you are working with real numbers or some other infinite field, you get infinitely many solutions.)
Things are different if you look at systems which are not homogeneous - then there is also a possibility that there is no solution.
If the solution is only x=0, then cx=0 is still 0. I do see your point that the theorem points out that the homogeneous system always at least has the solution x=0 whereas the inhomogeneous system may not have any solution at all.
– Bertram
Nov 22 at 18:44
add a comment |
up vote
0
down vote
From the given theorem we have that
- Does this imply that a homogeneous system $Ax=0$ has only a trivial solution or infinite number of solutions?
- recall that $det A=0$ or $det A neq 0$
- If not, then what values of the determinant for A imply that there is a non-trivial, unique solution, for a homogeneous equation?
- what about $det A neq 0$?
- If it does imply that $Ax=0$ only has a trivial or infinite (free parameter) solution, isn't that kind of weird?
- it is not weird, it is a theorem
Thank you for answering. The case of "what about det A not = 0" is addressed in the quote as leading to a trivial solution for a homogeneous system.
– Bertram
Nov 22 at 18:39
@Bertram The key fact is that if $det A neq 0$ and the columns $c_1,c_2,c_3$ of $A$ are linearly independent then $$Ax=x_1c_1+x_2c_2+x_3c_3=0 iff x=0$$
– gimusi
Nov 22 at 18:42
Yes thank you gimusi. I see now that the definition of the linear independence of the columns directly leads to the requirement that x=0 if it does solve the system of equations. Thank you.
– Bertram
Nov 23 at 17:12
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',
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%2f3009463%2fhomogeneous-only-trivial-or-infinite%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
It is good to notice that a homogeneous system $Ax=0$ always has at least zero solution $x=0$. (So for homogeneous system you cannot have no solutions at all.)
If you have any solution $x$, than any scalar multiple $cx$ is also a solution of the same homogeneous system. So once you have at least one non-zero solution, you immediately get many other solutions. (If you are working with real numbers or some other infinite field, you get infinitely many solutions.)
Things are different if you look at systems which are not homogeneous - then there is also a possibility that there is no solution.
If the solution is only x=0, then cx=0 is still 0. I do see your point that the theorem points out that the homogeneous system always at least has the solution x=0 whereas the inhomogeneous system may not have any solution at all.
– Bertram
Nov 22 at 18:44
add a comment |
up vote
1
down vote
It is good to notice that a homogeneous system $Ax=0$ always has at least zero solution $x=0$. (So for homogeneous system you cannot have no solutions at all.)
If you have any solution $x$, than any scalar multiple $cx$ is also a solution of the same homogeneous system. So once you have at least one non-zero solution, you immediately get many other solutions. (If you are working with real numbers or some other infinite field, you get infinitely many solutions.)
Things are different if you look at systems which are not homogeneous - then there is also a possibility that there is no solution.
If the solution is only x=0, then cx=0 is still 0. I do see your point that the theorem points out that the homogeneous system always at least has the solution x=0 whereas the inhomogeneous system may not have any solution at all.
– Bertram
Nov 22 at 18:44
add a comment |
up vote
1
down vote
up vote
1
down vote
It is good to notice that a homogeneous system $Ax=0$ always has at least zero solution $x=0$. (So for homogeneous system you cannot have no solutions at all.)
If you have any solution $x$, than any scalar multiple $cx$ is also a solution of the same homogeneous system. So once you have at least one non-zero solution, you immediately get many other solutions. (If you are working with real numbers or some other infinite field, you get infinitely many solutions.)
Things are different if you look at systems which are not homogeneous - then there is also a possibility that there is no solution.
It is good to notice that a homogeneous system $Ax=0$ always has at least zero solution $x=0$. (So for homogeneous system you cannot have no solutions at all.)
If you have any solution $x$, than any scalar multiple $cx$ is also a solution of the same homogeneous system. So once you have at least one non-zero solution, you immediately get many other solutions. (If you are working with real numbers or some other infinite field, you get infinitely many solutions.)
Things are different if you look at systems which are not homogeneous - then there is also a possibility that there is no solution.
answered Nov 22 at 18:26
Martin Sleziak
44.6k7115269
44.6k7115269
If the solution is only x=0, then cx=0 is still 0. I do see your point that the theorem points out that the homogeneous system always at least has the solution x=0 whereas the inhomogeneous system may not have any solution at all.
– Bertram
Nov 22 at 18:44
add a comment |
If the solution is only x=0, then cx=0 is still 0. I do see your point that the theorem points out that the homogeneous system always at least has the solution x=0 whereas the inhomogeneous system may not have any solution at all.
– Bertram
Nov 22 at 18:44
If the solution is only x=0, then cx=0 is still 0. I do see your point that the theorem points out that the homogeneous system always at least has the solution x=0 whereas the inhomogeneous system may not have any solution at all.
– Bertram
Nov 22 at 18:44
If the solution is only x=0, then cx=0 is still 0. I do see your point that the theorem points out that the homogeneous system always at least has the solution x=0 whereas the inhomogeneous system may not have any solution at all.
– Bertram
Nov 22 at 18:44
add a comment |
up vote
0
down vote
From the given theorem we have that
- Does this imply that a homogeneous system $Ax=0$ has only a trivial solution or infinite number of solutions?
- recall that $det A=0$ or $det A neq 0$
- If not, then what values of the determinant for A imply that there is a non-trivial, unique solution, for a homogeneous equation?
- what about $det A neq 0$?
- If it does imply that $Ax=0$ only has a trivial or infinite (free parameter) solution, isn't that kind of weird?
- it is not weird, it is a theorem
Thank you for answering. The case of "what about det A not = 0" is addressed in the quote as leading to a trivial solution for a homogeneous system.
– Bertram
Nov 22 at 18:39
@Bertram The key fact is that if $det A neq 0$ and the columns $c_1,c_2,c_3$ of $A$ are linearly independent then $$Ax=x_1c_1+x_2c_2+x_3c_3=0 iff x=0$$
– gimusi
Nov 22 at 18:42
Yes thank you gimusi. I see now that the definition of the linear independence of the columns directly leads to the requirement that x=0 if it does solve the system of equations. Thank you.
– Bertram
Nov 23 at 17:12
add a comment |
up vote
0
down vote
From the given theorem we have that
- Does this imply that a homogeneous system $Ax=0$ has only a trivial solution or infinite number of solutions?
- recall that $det A=0$ or $det A neq 0$
- If not, then what values of the determinant for A imply that there is a non-trivial, unique solution, for a homogeneous equation?
- what about $det A neq 0$?
- If it does imply that $Ax=0$ only has a trivial or infinite (free parameter) solution, isn't that kind of weird?
- it is not weird, it is a theorem
Thank you for answering. The case of "what about det A not = 0" is addressed in the quote as leading to a trivial solution for a homogeneous system.
– Bertram
Nov 22 at 18:39
@Bertram The key fact is that if $det A neq 0$ and the columns $c_1,c_2,c_3$ of $A$ are linearly independent then $$Ax=x_1c_1+x_2c_2+x_3c_3=0 iff x=0$$
– gimusi
Nov 22 at 18:42
Yes thank you gimusi. I see now that the definition of the linear independence of the columns directly leads to the requirement that x=0 if it does solve the system of equations. Thank you.
– Bertram
Nov 23 at 17:12
add a comment |
up vote
0
down vote
up vote
0
down vote
From the given theorem we have that
- Does this imply that a homogeneous system $Ax=0$ has only a trivial solution or infinite number of solutions?
- recall that $det A=0$ or $det A neq 0$
- If not, then what values of the determinant for A imply that there is a non-trivial, unique solution, for a homogeneous equation?
- what about $det A neq 0$?
- If it does imply that $Ax=0$ only has a trivial or infinite (free parameter) solution, isn't that kind of weird?
- it is not weird, it is a theorem
From the given theorem we have that
- Does this imply that a homogeneous system $Ax=0$ has only a trivial solution or infinite number of solutions?
- recall that $det A=0$ or $det A neq 0$
- If not, then what values of the determinant for A imply that there is a non-trivial, unique solution, for a homogeneous equation?
- what about $det A neq 0$?
- If it does imply that $Ax=0$ only has a trivial or infinite (free parameter) solution, isn't that kind of weird?
- it is not weird, it is a theorem
answered Nov 22 at 18:25
gimusi
92.7k94495
92.7k94495
Thank you for answering. The case of "what about det A not = 0" is addressed in the quote as leading to a trivial solution for a homogeneous system.
– Bertram
Nov 22 at 18:39
@Bertram The key fact is that if $det A neq 0$ and the columns $c_1,c_2,c_3$ of $A$ are linearly independent then $$Ax=x_1c_1+x_2c_2+x_3c_3=0 iff x=0$$
– gimusi
Nov 22 at 18:42
Yes thank you gimusi. I see now that the definition of the linear independence of the columns directly leads to the requirement that x=0 if it does solve the system of equations. Thank you.
– Bertram
Nov 23 at 17:12
add a comment |
Thank you for answering. The case of "what about det A not = 0" is addressed in the quote as leading to a trivial solution for a homogeneous system.
– Bertram
Nov 22 at 18:39
@Bertram The key fact is that if $det A neq 0$ and the columns $c_1,c_2,c_3$ of $A$ are linearly independent then $$Ax=x_1c_1+x_2c_2+x_3c_3=0 iff x=0$$
– gimusi
Nov 22 at 18:42
Yes thank you gimusi. I see now that the definition of the linear independence of the columns directly leads to the requirement that x=0 if it does solve the system of equations. Thank you.
– Bertram
Nov 23 at 17:12
Thank you for answering. The case of "what about det A not = 0" is addressed in the quote as leading to a trivial solution for a homogeneous system.
– Bertram
Nov 22 at 18:39
Thank you for answering. The case of "what about det A not = 0" is addressed in the quote as leading to a trivial solution for a homogeneous system.
– Bertram
Nov 22 at 18:39
@Bertram The key fact is that if $det A neq 0$ and the columns $c_1,c_2,c_3$ of $A$ are linearly independent then $$Ax=x_1c_1+x_2c_2+x_3c_3=0 iff x=0$$
– gimusi
Nov 22 at 18:42
@Bertram The key fact is that if $det A neq 0$ and the columns $c_1,c_2,c_3$ of $A$ are linearly independent then $$Ax=x_1c_1+x_2c_2+x_3c_3=0 iff x=0$$
– gimusi
Nov 22 at 18:42
Yes thank you gimusi. I see now that the definition of the linear independence of the columns directly leads to the requirement that x=0 if it does solve the system of equations. Thank you.
– Bertram
Nov 23 at 17:12
Yes thank you gimusi. I see now that the definition of the linear independence of the columns directly leads to the requirement that x=0 if it does solve the system of equations. Thank you.
– Bertram
Nov 23 at 17:12
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3009463%2fhomogeneous-only-trivial-or-infinite%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