Prove a given collection of operators is invariant on every subspace of a vector space
$begingroup$
I need to prove the following theorem and am quite lost, especially in proving the dimensions of the subspaces:
Let $V$ be a finite dimensional vector space. let $C_2$ be the collection of linear operators ${I, T}$ on $V$ such that $Tneq I$ and $T^2= I$. Then there is always a way to express $V$ as a direct sum of subspaces $V=U_1oplus U_2 oplus ...oplus U_m$ such that $C_2$ is invariant on each subspace. If our vector space is over the field $mathbb F=mathbb R $ then each subspace is of dimension at most 2.
linear-algebra abstract-algebra group-theory
asked Dec 3 '18 at 21:03
jmarsjmars
22
$endgroup$
add a comment |
$begingroup$
I need to prove the following theorem and am quite lost, especially in proving the dimensions of the subspaces:
Let $V$ be a finite dimensional vector space. let $C_2$ be the collection of linear operators ${I, T}$ on $V$ such that $Tneq I$ and $T^2= I$. Then there is always a way to express $V$ as a direct sum of subspaces $V=U_1oplus U_2 oplus ...oplus U_m$ such that $C_2$ is invariant on each subspace. If our vector space is over the field $mathbb F=mathbb R $ then each subspace is of dimension at most 2.
linear-algebra abstract-algebra group-theory
asked Dec 3 '18 at 21:03
jmarsjmars
22
$endgroup$
$begingroup$
Hint: If $vin V$, $vneq 0$, then either $T(v)inmathrm{span}(v)$ or $T(v)notin mathrm{span}(v)$. In the former case, is $mathrm{span}(v)$ invariant? In the latter case, what happens when you look at $mathrm{span}(v,T(v))$?
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:06
$begingroup$
In the former case, $span(v)$ is invariant. I'm not sure about the latter. I guess I'm confused as to how, if each subspace of the direct sum is unique, the subspaces can share the same elements.
$endgroup$
– jmars
Dec 3 '18 at 21:18
$begingroup$
They don’t; they intersect trivially; but what does that have to do with anything? In any case, you seem to have ignored one of your hypotheses, so use it...
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:22
$begingroup$
If I can show that $V=span(v)oplus span(v, T(v))$ then I've proved it?
$endgroup$
– jmars
Dec 3 '18 at 21:30
$begingroup$
If you can show that, you can show that Mathematics is inconsistent, because it’s false. That’s not the point. You are looking for invariant subspaces.
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:32
add a comment |
$begingroup$
I need to prove the following theorem and am quite lost, especially in proving the dimensions of the subspaces:
Let $V$ be a finite dimensional vector space. let $C_2$ be the collection of linear operators ${I, T}$ on $V$ such that $Tneq I$ and $T^2= I$. Then there is always a way to express $V$ as a direct sum of subspaces $V=U_1oplus U_2 oplus ...oplus U_m$ such that $C_2$ is invariant on each subspace. If our vector space is over the field $mathbb F=mathbb R $ then each subspace is of dimension at most 2.
linear-algebra abstract-algebra group-theory
asked Dec 3 '18 at 21:03
jmarsjmars
22
$endgroup$
I need to prove the following theorem and am quite lost, especially in proving the dimensions of the subspaces:
Let $V$ be a finite dimensional vector space. let $C_2$ be the collection of linear operators ${I, T}$ on $V$ such that $Tneq I$ and $T^2= I$. Then there is always a way to express $V$ as a direct sum of subspaces $V=U_1oplus U_2 oplus ...oplus U_m$ such that $C_2$ is invariant on each subspace. If our vector space is over the field $mathbb F=mathbb R $ then each subspace is of dimension at most 2.
linear-algebra abstract-algebra group-theory
linear-algebra abstract-algebra group-theory
asked Dec 3 '18 at 21:03
jmarsjmars
22
asked Dec 3 '18 at 21:03
jmarsjmars
22
asked Dec 3 '18 at 21:03
jmarsjmars
22
asked Dec 3 '18 at 21:03
jmarsjmars
22
asked Dec 3 '18 at 21:03
jmarsjmars
22
22
$begingroup$
Hint: If $vin V$, $vneq 0$, then either $T(v)inmathrm{span}(v)$ or $T(v)notin mathrm{span}(v)$. In the former case, is $mathrm{span}(v)$ invariant? In the latter case, what happens when you look at $mathrm{span}(v,T(v))$?
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:06
$begingroup$
In the former case, $span(v)$ is invariant. I'm not sure about the latter. I guess I'm confused as to how, if each subspace of the direct sum is unique, the subspaces can share the same elements.
$endgroup$
– jmars
Dec 3 '18 at 21:18
$begingroup$
They don’t; they intersect trivially; but what does that have to do with anything? In any case, you seem to have ignored one of your hypotheses, so use it...
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:22
$begingroup$
If I can show that $V=span(v)oplus span(v, T(v))$ then I've proved it?
$endgroup$
– jmars
Dec 3 '18 at 21:30
$begingroup$
If you can show that, you can show that Mathematics is inconsistent, because it’s false. That’s not the point. You are looking for invariant subspaces.
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:32
add a comment |
$begingroup$
Hint: If $vin V$, $vneq 0$, then either $T(v)inmathrm{span}(v)$ or $T(v)notin mathrm{span}(v)$. In the former case, is $mathrm{span}(v)$ invariant? In the latter case, what happens when you look at $mathrm{span}(v,T(v))$?
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:06
$begingroup$
In the former case, $span(v)$ is invariant. I'm not sure about the latter. I guess I'm confused as to how, if each subspace of the direct sum is unique, the subspaces can share the same elements.
$endgroup$
– jmars
Dec 3 '18 at 21:18
$begingroup$
They don’t; they intersect trivially; but what does that have to do with anything? In any case, you seem to have ignored one of your hypotheses, so use it...
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:22
$begingroup$
If I can show that $V=span(v)oplus span(v, T(v))$ then I've proved it?
$endgroup$
– jmars
Dec 3 '18 at 21:30
$begingroup$
If you can show that, you can show that Mathematics is inconsistent, because it’s false. That’s not the point. You are looking for invariant subspaces.
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:32
$begingroup$
Hint: If $vin V$, $vneq 0$, then either $T(v)inmathrm{span}(v)$ or $T(v)notin mathrm{span}(v)$. In the former case, is $mathrm{span}(v)$ invariant? In the latter case, what happens when you look at $mathrm{span}(v,T(v))$?
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:06
$begingroup$
Hint: If $vin V$, $vneq 0$, then either $T(v)inmathrm{span}(v)$ or $T(v)notin mathrm{span}(v)$. In the former case, is $mathrm{span}(v)$ invariant? In the latter case, what happens when you look at $mathrm{span}(v,T(v))$?
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:06
$begingroup$
In the former case, $span(v)$ is invariant. I'm not sure about the latter. I guess I'm confused as to how, if each subspace of the direct sum is unique, the subspaces can share the same elements.
$endgroup$
– jmars
Dec 3 '18 at 21:18
$begingroup$
In the former case, $span(v)$ is invariant. I'm not sure about the latter. I guess I'm confused as to how, if each subspace of the direct sum is unique, the subspaces can share the same elements.
$endgroup$
– jmars
Dec 3 '18 at 21:18
$begingroup$
They don’t; they intersect trivially; but what does that have to do with anything? In any case, you seem to have ignored one of your hypotheses, so use it...
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:22
$begingroup$
They don’t; they intersect trivially; but what does that have to do with anything? In any case, you seem to have ignored one of your hypotheses, so use it...
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:22
$begingroup$
If I can show that $V=span(v)oplus span(v, T(v))$ then I've proved it?
$endgroup$
– jmars
Dec 3 '18 at 21:30
$begingroup$
If I can show that $V=span(v)oplus span(v, T(v))$ then I've proved it?
$endgroup$
– jmars
Dec 3 '18 at 21:30
$begingroup$
If you can show that, you can show that Mathematics is inconsistent, because it’s false. That’s not the point. You are looking for invariant subspaces.
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:32
$begingroup$
If you can show that, you can show that Mathematics is inconsistent, because it’s false. That’s not the point. You are looking for invariant subspaces.
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:32
add a comment |
0
active
oldest
votes
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%2f3024661%2fprove-a-given-collection-of-operators-is-invariant-on-every-subspace-of-a-vector%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3024661%2fprove-a-given-collection-of-operators-is-invariant-on-every-subspace-of-a-vector%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
$begingroup$
Hint: If $vin V$, $vneq 0$, then either $T(v)inmathrm{span}(v)$ or $T(v)notin mathrm{span}(v)$. In the former case, is $mathrm{span}(v)$ invariant? In the latter case, what happens when you look at $mathrm{span}(v,T(v))$?
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:06
$begingroup$
In the former case, $span(v)$ is invariant. I'm not sure about the latter. I guess I'm confused as to how, if each subspace of the direct sum is unique, the subspaces can share the same elements.
$endgroup$
– jmars
Dec 3 '18 at 21:18
$begingroup$
They don’t; they intersect trivially; but what does that have to do with anything? In any case, you seem to have ignored one of your hypotheses, so use it...
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:22
$begingroup$
If I can show that $V=span(v)oplus span(v, T(v))$ then I've proved it?
$endgroup$
– jmars
Dec 3 '18 at 21:30
$begingroup$
If you can show that, you can show that Mathematics is inconsistent, because it’s false. That’s not the point. You are looking for invariant subspaces.
$endgroup$
– Arturo Magidin
Dec 3 '18 at 21:32