How to calculate subspace of a set of solutions of matrix Ax=b
0
$begingroup$
I am looking through some old linear algebra exam papers. However i do not understand how to calculate whether a set of solutions is within a certain subspace R. This is the problem in question:
I think i understand how to check whether vectors are within a subspace R, but how would i calculate this?
Thanks a lot, really hope you can help me out!
linear-algebra matrices vector-spaces invariant-subspace
edited Dec 28 '16 at 18:59
Dan Rust
22.8k114884
asked Dec 28 '16 at 18:54
user102937user102937
257
$endgroup$
|
show 1 more comment
0
$begingroup$
I am looking through some old linear algebra exam papers. However i do not understand how to calculate whether a set of solutions is within a certain subspace R. This is the problem in question:
I think i understand how to check whether vectors are within a subspace R, but how would i calculate this?
Thanks a lot, really hope you can help me out!
linear-algebra matrices vector-spaces invariant-subspace
edited Dec 28 '16 at 18:59
Dan Rust
22.8k114884
asked Dec 28 '16 at 18:54
user102937user102937
257
$endgroup$
1
$begingroup$
What is $mathcal{R}$?
$endgroup$
– Dan Rust
Dec 28 '16 at 19:04
$begingroup$
What have you tried? What is your understanding of how to determine if a set is a subspace of the vector space $mathbb{R} $?
$endgroup$
– user23793
Dec 28 '16 at 19:04
$begingroup$
I tried to follow this video:
$endgroup$
– user102937
Dec 28 '16 at 19:09
$begingroup$
I tried to follow this video: youtube.com/watch?v=q97HmMdD8ZM And to me it seems that the subspace is equal to the amount of rows of the vectors within the span. Am i completely off track or?
$endgroup$
– user102937
Dec 28 '16 at 19:10
$begingroup$
I don't understand the question. What do you want to calculate?
$endgroup$
– Jack
Dec 28 '16 at 19:47
|
show 1 more comment
0
0
0
$begingroup$
I am looking through some old linear algebra exam papers. However i do not understand how to calculate whether a set of solutions is within a certain subspace R. This is the problem in question:
I think i understand how to check whether vectors are within a subspace R, but how would i calculate this?
Thanks a lot, really hope you can help me out!
linear-algebra matrices vector-spaces invariant-subspace
edited Dec 28 '16 at 18:59
Dan Rust
22.8k114884
asked Dec 28 '16 at 18:54
user102937user102937
257
$endgroup$
I am looking through some old linear algebra exam papers. However i do not understand how to calculate whether a set of solutions is within a certain subspace R. This is the problem in question:
I think i understand how to check whether vectors are within a subspace R, but how would i calculate this?
Thanks a lot, really hope you can help me out!
linear-algebra matrices vector-spaces invariant-subspace
linear-algebra matrices vector-spaces invariant-subspace
edited Dec 28 '16 at 18:59
Dan Rust
22.8k114884
asked Dec 28 '16 at 18:54
user102937user102937
257
edited Dec 28 '16 at 18:59
Dan Rust
22.8k114884
asked Dec 28 '16 at 18:54
user102937user102937
257
edited Dec 28 '16 at 18:59
Dan Rust
22.8k114884
edited Dec 28 '16 at 18:59
Dan Rust
22.8k114884
edited Dec 28 '16 at 18:59
Dan Rust
22.8k114884
22.8k114884
asked Dec 28 '16 at 18:54
user102937user102937
257
asked Dec 28 '16 at 18:54
user102937user102937
257
asked Dec 28 '16 at 18:54
user102937user102937
257
257
1
$begingroup$
What is $mathcal{R}$?
$endgroup$
– Dan Rust
Dec 28 '16 at 19:04
$begingroup$
What have you tried? What is your understanding of how to determine if a set is a subspace of the vector space $mathbb{R} $?
$endgroup$
– user23793
Dec 28 '16 at 19:04
$begingroup$
I tried to follow this video:
$endgroup$
– user102937
Dec 28 '16 at 19:09
$begingroup$
I tried to follow this video: youtube.com/watch?v=q97HmMdD8ZM And to me it seems that the subspace is equal to the amount of rows of the vectors within the span. Am i completely off track or?
$endgroup$
– user102937
Dec 28 '16 at 19:10
$begingroup$
I don't understand the question. What do you want to calculate?
$endgroup$
– Jack
Dec 28 '16 at 19:47
|
show 1 more comment
1
$begingroup$
What is $mathcal{R}$?
$endgroup$
– Dan Rust
Dec 28 '16 at 19:04
$begingroup$
What have you tried? What is your understanding of how to determine if a set is a subspace of the vector space $mathbb{R} $?
$endgroup$
– user23793
Dec 28 '16 at 19:04
$begingroup$
I tried to follow this video:
$endgroup$
– user102937
Dec 28 '16 at 19:09
$begingroup$
I tried to follow this video: youtube.com/watch?v=q97HmMdD8ZM And to me it seems that the subspace is equal to the amount of rows of the vectors within the span. Am i completely off track or?
$endgroup$
– user102937
Dec 28 '16 at 19:10
$begingroup$
I don't understand the question. What do you want to calculate?
$endgroup$
– Jack
Dec 28 '16 at 19:47
1
1
$begingroup$
What is $mathcal{R}$?
$endgroup$
– Dan Rust
Dec 28 '16 at 19:04
$begingroup$
What is $mathcal{R}$?
$endgroup$
– Dan Rust
Dec 28 '16 at 19:04
$begingroup$
What have you tried? What is your understanding of how to determine if a set is a subspace of the vector space $mathbb{R} $?
$endgroup$
– user23793
Dec 28 '16 at 19:04
$begingroup$
What have you tried? What is your understanding of how to determine if a set is a subspace of the vector space $mathbb{R} $?
$endgroup$
– user23793
Dec 28 '16 at 19:04
$begingroup$
I tried to follow this video:
$endgroup$
– user102937
Dec 28 '16 at 19:09
$begingroup$
I tried to follow this video:
$endgroup$
– user102937
Dec 28 '16 at 19:09
$begingroup$
I tried to follow this video: youtube.com/watch?v=q97HmMdD8ZM And to me it seems that the subspace is equal to the amount of rows of the vectors within the span. Am i completely off track or?
$endgroup$
– user102937
Dec 28 '16 at 19:10
$begingroup$
I tried to follow this video: youtube.com/watch?v=q97HmMdD8ZM And to me it seems that the subspace is equal to the amount of rows of the vectors within the span. Am i completely off track or?
$endgroup$
– user102937
Dec 28 '16 at 19:10
$begingroup$
I don't understand the question. What do you want to calculate?
$endgroup$
– Jack
Dec 28 '16 at 19:47
$begingroup$
I don't understand the question. What do you want to calculate?
$endgroup$
– Jack
Dec 28 '16 at 19:47
|
show 1 more comment
1 Answer
1
active
oldest
votes
1
$begingroup$
The set of solutions of $Amathbf{x}=mathbf{b}$ is not a subespace of $mathbb{R}^4$ becasuse the null vector $mathbf{0}=begin{bmatrix}{0}\{0}\{0}\{0}end{bmatrix}$ does not satisfy $Amathbf{0}=mathbf{b}$.
answered Dec 28 '16 at 19:11
Fernando RevillaFernando Revilla
3,322520
$endgroup$
$begingroup$
Thank you. Is this the same as saying that it is because the last row does not equal the zero vector?
$endgroup$
– user102937
Dec 28 '16 at 19:14
$begingroup$
- Of the RREF of Ax=B, i mean. Sorry i do not understand how to edit my comments.
$endgroup$
– user102937
Dec 28 '16 at 19:15
$begingroup$
No, nothing to do. For example, $Amathbf{x}=mathbf{0}$ would be a subspace of $mathbb{R}^4$ with $A$ the same matrix.
$endgroup$
– Fernando Revilla
Dec 28 '16 at 19:17
$begingroup$
@user102937 No, it’s the same as saying that $mathbf bnemathbf 0$. The solution set to $Amathbf x=mathbf b$ can only be a vector space when $mathbf b=mathbf 0$. The rref in the problem is sort of a red herring.
$endgroup$
– amd
Dec 28 '16 at 19:22
$begingroup$
Thanks again Alone and amd. So just to be completely sure i get it: If and only if b=0, can said problem be true? And if b=0 is it then always within the subspace R^n?
$endgroup$
– user102937
Dec 28 '16 at 20:16
|
show 1 more 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
});
}
});
draft saved
draft discarded
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%2f2075141%2fhow-to-calculate-subspace-of-a-set-of-solutions-of-matrix-ax-b%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
1
$begingroup$
The set of solutions of $Amathbf{x}=mathbf{b}$ is not a subespace of $mathbb{R}^4$ becasuse the null vector $mathbf{0}=begin{bmatrix}{0}\{0}\{0}\{0}end{bmatrix}$ does not satisfy $Amathbf{0}=mathbf{b}$.
answered Dec 28 '16 at 19:11
Fernando RevillaFernando Revilla
3,322520
$endgroup$
$begingroup$
Thank you. Is this the same as saying that it is because the last row does not equal the zero vector?
$endgroup$
– user102937
Dec 28 '16 at 19:14
$begingroup$
- Of the RREF of Ax=B, i mean. Sorry i do not understand how to edit my comments.
$endgroup$
– user102937
Dec 28 '16 at 19:15
$begingroup$
No, nothing to do. For example, $Amathbf{x}=mathbf{0}$ would be a subspace of $mathbb{R}^4$ with $A$ the same matrix.
$endgroup$
– Fernando Revilla
Dec 28 '16 at 19:17
$begingroup$
@user102937 No, it’s the same as saying that $mathbf bnemathbf 0$. The solution set to $Amathbf x=mathbf b$ can only be a vector space when $mathbf b=mathbf 0$. The rref in the problem is sort of a red herring.
$endgroup$
– amd
Dec 28 '16 at 19:22
$begingroup$
Thanks again Alone and amd. So just to be completely sure i get it: If and only if b=0, can said problem be true? And if b=0 is it then always within the subspace R^n?
$endgroup$
– user102937
Dec 28 '16 at 20:16
|
show 1 more comment
1
$begingroup$
The set of solutions of $Amathbf{x}=mathbf{b}$ is not a subespace of $mathbb{R}^4$ becasuse the null vector $mathbf{0}=begin{bmatrix}{0}\{0}\{0}\{0}end{bmatrix}$ does not satisfy $Amathbf{0}=mathbf{b}$.
answered Dec 28 '16 at 19:11
Fernando RevillaFernando Revilla
3,322520
$endgroup$
$begingroup$
Thank you. Is this the same as saying that it is because the last row does not equal the zero vector?
$endgroup$
– user102937
Dec 28 '16 at 19:14
$begingroup$
- Of the RREF of Ax=B, i mean. Sorry i do not understand how to edit my comments.
$endgroup$
– user102937
Dec 28 '16 at 19:15
$begingroup$
No, nothing to do. For example, $Amathbf{x}=mathbf{0}$ would be a subspace of $mathbb{R}^4$ with $A$ the same matrix.
$endgroup$
– Fernando Revilla
Dec 28 '16 at 19:17
$begingroup$
@user102937 No, it’s the same as saying that $mathbf bnemathbf 0$. The solution set to $Amathbf x=mathbf b$ can only be a vector space when $mathbf b=mathbf 0$. The rref in the problem is sort of a red herring.
$endgroup$
– amd
Dec 28 '16 at 19:22
$begingroup$
Thanks again Alone and amd. So just to be completely sure i get it: If and only if b=0, can said problem be true? And if b=0 is it then always within the subspace R^n?
$endgroup$
– user102937
Dec 28 '16 at 20:16
|
show 1 more comment
1
1
1
$begingroup$
The set of solutions of $Amathbf{x}=mathbf{b}$ is not a subespace of $mathbb{R}^4$ becasuse the null vector $mathbf{0}=begin{bmatrix}{0}\{0}\{0}\{0}end{bmatrix}$ does not satisfy $Amathbf{0}=mathbf{b}$.
answered Dec 28 '16 at 19:11
Fernando RevillaFernando Revilla
3,322520
$endgroup$
The set of solutions of $Amathbf{x}=mathbf{b}$ is not a subespace of $mathbb{R}^4$ becasuse the null vector $mathbf{0}=begin{bmatrix}{0}\{0}\{0}\{0}end{bmatrix}$ does not satisfy $Amathbf{0}=mathbf{b}$.
answered Dec 28 '16 at 19:11
Fernando RevillaFernando Revilla
3,322520
answered Dec 28 '16 at 19:11
Fernando RevillaFernando Revilla
3,322520
answered Dec 28 '16 at 19:11
Fernando RevillaFernando Revilla
3,322520
answered Dec 28 '16 at 19:11
Fernando RevillaFernando Revilla
3,322520
3,322520
$begingroup$
Thank you. Is this the same as saying that it is because the last row does not equal the zero vector?
$endgroup$
– user102937
Dec 28 '16 at 19:14
$begingroup$
- Of the RREF of Ax=B, i mean. Sorry i do not understand how to edit my comments.
$endgroup$
– user102937
Dec 28 '16 at 19:15
$begingroup$
No, nothing to do. For example, $Amathbf{x}=mathbf{0}$ would be a subspace of $mathbb{R}^4$ with $A$ the same matrix.
$endgroup$
– Fernando Revilla
Dec 28 '16 at 19:17
$begingroup$
@user102937 No, it’s the same as saying that $mathbf bnemathbf 0$. The solution set to $Amathbf x=mathbf b$ can only be a vector space when $mathbf b=mathbf 0$. The rref in the problem is sort of a red herring.
$endgroup$
– amd
Dec 28 '16 at 19:22
$begingroup$
Thanks again Alone and amd. So just to be completely sure i get it: If and only if b=0, can said problem be true? And if b=0 is it then always within the subspace R^n?
$endgroup$
– user102937
Dec 28 '16 at 20:16
|
show 1 more comment
$begingroup$
Thank you. Is this the same as saying that it is because the last row does not equal the zero vector?
$endgroup$
– user102937
Dec 28 '16 at 19:14
$begingroup$
- Of the RREF of Ax=B, i mean. Sorry i do not understand how to edit my comments.
$endgroup$
– user102937
Dec 28 '16 at 19:15
$begingroup$
No, nothing to do. For example, $Amathbf{x}=mathbf{0}$ would be a subspace of $mathbb{R}^4$ with $A$ the same matrix.
$endgroup$
– Fernando Revilla
Dec 28 '16 at 19:17
$begingroup$
@user102937 No, it’s the same as saying that $mathbf bnemathbf 0$. The solution set to $Amathbf x=mathbf b$ can only be a vector space when $mathbf b=mathbf 0$. The rref in the problem is sort of a red herring.
$endgroup$
– amd
Dec 28 '16 at 19:22
$begingroup$
Thanks again Alone and amd. So just to be completely sure i get it: If and only if b=0, can said problem be true? And if b=0 is it then always within the subspace R^n?
$endgroup$
– user102937
Dec 28 '16 at 20:16
$begingroup$
Thank you. Is this the same as saying that it is because the last row does not equal the zero vector?
$endgroup$
– user102937
Dec 28 '16 at 19:14
$begingroup$
Thank you. Is this the same as saying that it is because the last row does not equal the zero vector?
$endgroup$
– user102937
Dec 28 '16 at 19:14
$begingroup$
- Of the RREF of Ax=B, i mean. Sorry i do not understand how to edit my comments.
$endgroup$
– user102937
Dec 28 '16 at 19:15
$begingroup$
- Of the RREF of Ax=B, i mean. Sorry i do not understand how to edit my comments.
$endgroup$
– user102937
Dec 28 '16 at 19:15
$begingroup$
No, nothing to do. For example, $Amathbf{x}=mathbf{0}$ would be a subspace of $mathbb{R}^4$ with $A$ the same matrix.
$endgroup$
– Fernando Revilla
Dec 28 '16 at 19:17
$begingroup$
No, nothing to do. For example, $Amathbf{x}=mathbf{0}$ would be a subspace of $mathbb{R}^4$ with $A$ the same matrix.
$endgroup$
– Fernando Revilla
Dec 28 '16 at 19:17
$begingroup$
@user102937 No, it’s the same as saying that $mathbf bnemathbf 0$. The solution set to $Amathbf x=mathbf b$ can only be a vector space when $mathbf b=mathbf 0$. The rref in the problem is sort of a red herring.
$endgroup$
– amd
Dec 28 '16 at 19:22
$begingroup$
@user102937 No, it’s the same as saying that $mathbf bnemathbf 0$. The solution set to $Amathbf x=mathbf b$ can only be a vector space when $mathbf b=mathbf 0$. The rref in the problem is sort of a red herring.
$endgroup$
– amd
Dec 28 '16 at 19:22
$begingroup$
Thanks again Alone and amd. So just to be completely sure i get it: If and only if b=0, can said problem be true? And if b=0 is it then always within the subspace R^n?
$endgroup$
– user102937
Dec 28 '16 at 20:16
$begingroup$
Thanks again Alone and amd. So just to be completely sure i get it: If and only if b=0, can said problem be true? And if b=0 is it then always within the subspace R^n?
$endgroup$
– user102937
Dec 28 '16 at 20:16
|
show 1 more comment
draft saved
draft discarded
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.
draft saved
draft discarded
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%2f2075141%2fhow-to-calculate-subspace-of-a-set-of-solutions-of-matrix-ax-b%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
1
$begingroup$
What is $mathcal{R}$?
$endgroup$
– Dan Rust
Dec 28 '16 at 19:04
$begingroup$
What have you tried? What is your understanding of how to determine if a set is a subspace of the vector space $mathbb{R} $?
$endgroup$
– user23793
Dec 28 '16 at 19:04
$begingroup$
I tried to follow this video:
$endgroup$
– user102937
Dec 28 '16 at 19:09
$begingroup$
I tried to follow this video: youtube.com/watch?v=q97HmMdD8ZM And to me it seems that the subspace is equal to the amount of rows of the vectors within the span. Am i completely off track or?
$endgroup$
– user102937
Dec 28 '16 at 19:10
$begingroup$
I don't understand the question. What do you want to calculate?
$endgroup$
– Jack
Dec 28 '16 at 19:47