Linear Algebra: Dimension of kernel
$begingroup$
Suppose that we have the vector space
$V={f/f:Rrightarrow R, text{every class derivative is defined in R}}$
and $φ: Vrightarrow W$ with $φ(f)=f+f'$ is linear.
I want to find the dimension of the $Ker φ$ and if $Βleq V$ with $Βcap A={0}$ to show that the restriction of $φ$ in $B$ is one to one.
Any ideas please?
linear-algebra vector-fields
asked Jan 8 at 8:20
GeorgeGeorge
32
$endgroup$
migrated from stats.stackexchange.com Jan 8 at 11:22
This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.
add a comment |
$begingroup$
Suppose that we have the vector space
$V={f/f:Rrightarrow R, text{every class derivative is defined in R}}$
and $φ: Vrightarrow W$ with $φ(f)=f+f'$ is linear.
I want to find the dimension of the $Ker φ$ and if $Βleq V$ with $Βcap A={0}$ to show that the restriction of $φ$ in $B$ is one to one.
Any ideas please?
linear-algebra vector-fields
asked Jan 8 at 8:20
GeorgeGeorge
32
$endgroup$
migrated from stats.stackexchange.com Jan 8 at 11:22
This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.
$begingroup$
what's a class derivative
$endgroup$
– seanv507
Jan 8 at 10:14
add a comment |
$begingroup$
Suppose that we have the vector space
$V={f/f:Rrightarrow R, text{every class derivative is defined in R}}$
and $φ: Vrightarrow W$ with $φ(f)=f+f'$ is linear.
I want to find the dimension of the $Ker φ$ and if $Βleq V$ with $Βcap A={0}$ to show that the restriction of $φ$ in $B$ is one to one.
Any ideas please?
linear-algebra vector-fields
asked Jan 8 at 8:20
GeorgeGeorge
32
$endgroup$
Suppose that we have the vector space
$V={f/f:Rrightarrow R, text{every class derivative is defined in R}}$
and $φ: Vrightarrow W$ with $φ(f)=f+f'$ is linear.
I want to find the dimension of the $Ker φ$ and if $Βleq V$ with $Βcap A={0}$ to show that the restriction of $φ$ in $B$ is one to one.
Any ideas please?
linear-algebra vector-fields
linear-algebra vector-fields
asked Jan 8 at 8:20
GeorgeGeorge
32
asked Jan 8 at 8:20
GeorgeGeorge
32
asked Jan 8 at 8:20
GeorgeGeorge
32
asked Jan 8 at 8:20
GeorgeGeorge
32
asked Jan 8 at 8:20
GeorgeGeorge
32
32
migrated from stats.stackexchange.com Jan 8 at 11:22
This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.
migrated from stats.stackexchange.com Jan 8 at 11:22
This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.
$begingroup$
what's a class derivative
$endgroup$
– seanv507
Jan 8 at 10:14
add a comment |
$begingroup$
what's a class derivative
$endgroup$
– seanv507
Jan 8 at 10:14
$begingroup$
what's a class derivative
$endgroup$
– seanv507
Jan 8 at 10:14
$begingroup$
what's a class derivative
$endgroup$
– seanv507
Jan 8 at 10:14
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Let's $f in V$ and $g in text{Ker}{varphi} subset V$, thus we have $varphi(f+g) = varphi(f)$
So we can write:
$varphi(f+g) = varphi(f) + varphi(g) = f +f' + g +g' = f +f' Rightarrow g +g' = 0$
Which is an ODE with solution $g = ke^{-t}$. This spans the kernel which has dimension 1.
answered Jan 8 at 11:20
Carlos CamposCarlos Campos
58439
$endgroup$
add a comment |
$begingroup$
I agree with your solution. How can I prove now that the restriction of $phi$ in $B$ is one to one?
answered Jan 11 at 7:40
GeorgeGeorge
32
$endgroup$
add a comment |
Your Answer
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%2f3066050%2flinear-algebra-dimension-of-kernel%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
$begingroup$
Let's $f in V$ and $g in text{Ker}{varphi} subset V$, thus we have $varphi(f+g) = varphi(f)$
So we can write:
$varphi(f+g) = varphi(f) + varphi(g) = f +f' + g +g' = f +f' Rightarrow g +g' = 0$
Which is an ODE with solution $g = ke^{-t}$. This spans the kernel which has dimension 1.
answered Jan 8 at 11:20
Carlos CamposCarlos Campos
58439
$endgroup$
add a comment |
$begingroup$
Let's $f in V$ and $g in text{Ker}{varphi} subset V$, thus we have $varphi(f+g) = varphi(f)$
So we can write:
$varphi(f+g) = varphi(f) + varphi(g) = f +f' + g +g' = f +f' Rightarrow g +g' = 0$
Which is an ODE with solution $g = ke^{-t}$. This spans the kernel which has dimension 1.
answered Jan 8 at 11:20
Carlos CamposCarlos Campos
58439
$endgroup$
add a comment |
$begingroup$
Let's $f in V$ and $g in text{Ker}{varphi} subset V$, thus we have $varphi(f+g) = varphi(f)$
So we can write:
$varphi(f+g) = varphi(f) + varphi(g) = f +f' + g +g' = f +f' Rightarrow g +g' = 0$
Which is an ODE with solution $g = ke^{-t}$. This spans the kernel which has dimension 1.
answered Jan 8 at 11:20
Carlos CamposCarlos Campos
58439
$endgroup$
Let's $f in V$ and $g in text{Ker}{varphi} subset V$, thus we have $varphi(f+g) = varphi(f)$
So we can write:
$varphi(f+g) = varphi(f) + varphi(g) = f +f' + g +g' = f +f' Rightarrow g +g' = 0$
Which is an ODE with solution $g = ke^{-t}$. This spans the kernel which has dimension 1.
answered Jan 8 at 11:20
Carlos CamposCarlos Campos
58439
answered Jan 8 at 11:20
Carlos CamposCarlos Campos
58439
answered Jan 8 at 11:20
Carlos CamposCarlos Campos
58439
answered Jan 8 at 11:20
Carlos CamposCarlos Campos
58439
58439
add a comment |
add a comment |
$begingroup$
I agree with your solution. How can I prove now that the restriction of $phi$ in $B$ is one to one?
answered Jan 11 at 7:40
GeorgeGeorge
32
$endgroup$
add a comment |
$begingroup$
I agree with your solution. How can I prove now that the restriction of $phi$ in $B$ is one to one?
answered Jan 11 at 7:40
GeorgeGeorge
32
$endgroup$
add a comment |
$begingroup$
I agree with your solution. How can I prove now that the restriction of $phi$ in $B$ is one to one?
answered Jan 11 at 7:40
GeorgeGeorge
32
$endgroup$
I agree with your solution. How can I prove now that the restriction of $phi$ in $B$ is one to one?
answered Jan 11 at 7:40
GeorgeGeorge
32
answered Jan 11 at 7:40
GeorgeGeorge
32
answered Jan 11 at 7:40
GeorgeGeorge
32
answered Jan 11 at 7:40
GeorgeGeorge
32
32
add a comment |
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%2f3066050%2flinear-algebra-dimension-of-kernel%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
$begingroup$
what's a class derivative
$endgroup$
– seanv507
Jan 8 at 10:14