How do I work out the generator for $GL(2,Bbb{R})$?












2












$begingroup$


Where $G$ is a group $$G=GL(2,mathbb{R})$$



$$x=begin{bmatrix}
cos(pi/4) & sin(pi/4)\
-sin(pi/4)& cos(pi/4)
end{bmatrix}$$



How do I work out the generator $langle xrangle$... Quite new to group theory so any help would help, thanks.






linear-algebra group-theory trigonometry







share|cite|improve this question











$endgroup$












  • $begingroup$
    That's a rotation matrix. What are its powers?
    $endgroup$
    – Lord Shark the Unknown
    Dec 29 '18 at 11:33










  • $begingroup$
    What do you mean by rotation matrix?
    $endgroup$
    – Reety
    Dec 29 '18 at 11:33












  • $begingroup$
    In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. To learn more: en.wikipedia.org/wiki/Rotation_matrix
    $endgroup$
    – toric_actions
    Dec 29 '18 at 11:44
















2












$begingroup$


Where $G$ is a group $$G=GL(2,mathbb{R})$$



$$x=begin{bmatrix}
cos(pi/4) & sin(pi/4)\
-sin(pi/4)& cos(pi/4)
end{bmatrix}$$



How do I work out the generator $langle xrangle$... Quite new to group theory so any help would help, thanks.






linear-algebra group-theory trigonometry







share|cite|improve this question











$endgroup$












  • $begingroup$
    That's a rotation matrix. What are its powers?
    $endgroup$
    – Lord Shark the Unknown
    Dec 29 '18 at 11:33










  • $begingroup$
    What do you mean by rotation matrix?
    $endgroup$
    – Reety
    Dec 29 '18 at 11:33












  • $begingroup$
    In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. To learn more: en.wikipedia.org/wiki/Rotation_matrix
    $endgroup$
    – toric_actions
    Dec 29 '18 at 11:44














2












2








2


1



$begingroup$


Where $G$ is a group $$G=GL(2,mathbb{R})$$



$$x=begin{bmatrix}
cos(pi/4) & sin(pi/4)\
-sin(pi/4)& cos(pi/4)
end{bmatrix}$$



How do I work out the generator $langle xrangle$... Quite new to group theory so any help would help, thanks.






linear-algebra group-theory trigonometry







share|cite|improve this question











$endgroup$




Where $G$ is a group $$G=GL(2,mathbb{R})$$



$$x=begin{bmatrix}
cos(pi/4) & sin(pi/4)\
-sin(pi/4)& cos(pi/4)
end{bmatrix}$$



How do I work out the generator $langle xrangle$... Quite new to group theory so any help would help, thanks.






linear-algebra group-theory trigonometry




linear-algebra group-theory trigonometry






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 29 '18 at 12:11









mechanodroid

27.8k62447




27.8k62447










asked Dec 29 '18 at 11:31









ReetyReety

15311




15311












  • $begingroup$
    That's a rotation matrix. What are its powers?
    $endgroup$
    – Lord Shark the Unknown
    Dec 29 '18 at 11:33










  • $begingroup$
    What do you mean by rotation matrix?
    $endgroup$
    – Reety
    Dec 29 '18 at 11:33












  • $begingroup$
    In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. To learn more: en.wikipedia.org/wiki/Rotation_matrix
    $endgroup$
    – toric_actions
    Dec 29 '18 at 11:44


















  • $begingroup$
    That's a rotation matrix. What are its powers?
    $endgroup$
    – Lord Shark the Unknown
    Dec 29 '18 at 11:33










  • $begingroup$
    What do you mean by rotation matrix?
    $endgroup$
    – Reety
    Dec 29 '18 at 11:33












  • $begingroup$
    In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. To learn more: en.wikipedia.org/wiki/Rotation_matrix
    $endgroup$
    – toric_actions
    Dec 29 '18 at 11:44
















$begingroup$
That's a rotation matrix. What are its powers?
$endgroup$
– Lord Shark the Unknown
Dec 29 '18 at 11:33




$begingroup$
That's a rotation matrix. What are its powers?
$endgroup$
– Lord Shark the Unknown
Dec 29 '18 at 11:33












$begingroup$
What do you mean by rotation matrix?
$endgroup$
– Reety
Dec 29 '18 at 11:33






$begingroup$
What do you mean by rotation matrix?
$endgroup$
– Reety
Dec 29 '18 at 11:33














$begingroup$
In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. To learn more: en.wikipedia.org/wiki/Rotation_matrix
$endgroup$
– toric_actions
Dec 29 '18 at 11:44




$begingroup$
In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. To learn more: en.wikipedia.org/wiki/Rotation_matrix
$endgroup$
– toric_actions
Dec 29 '18 at 11:44










1 Answer
1






active

oldest

votes


















3












$begingroup$

Your matrix $x$ represents a rotation in $mathbb{R}^2$ around the origin by the angle $frac{pi}4$ clockwise.



In general $R(phi) = begin{bmatrix} cosphi & sinphi \ -sinphi & cosphiend{bmatrix}$ does the same thing with the angle $phi$.



Clearly if you rotate by $phi$ and then by $psi$, it is the same as rotating by $phi + psi$. Hence $R(phi)R(psi) = R(phi + psi)$. You can also check this algebraically, of course.



Also $R(phi)^{-1} = R(-phi)$ as the inverse of a rotation by $phi$ is a rotation by $-phi$.



Therefore



$$langle xrangle = {x^n : n in mathbb{Z}} = left{Rleft(frac{pi}4right)^n : n in mathbb{Z}right} = left{Rleft(frac{npi}4right) : n in mathbb{Z}right}$$



Hence your group consists of all rotations by an angle which is a multiple of $fracpi4$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This is a great explanation thank you, although the book I'm reading says the answer is {e, x, x^2, x^3} how can I convert this notation? - @mechanodroid
    $endgroup$
    – Reety
    Dec 29 '18 at 12:16












  • $begingroup$
    @Reety Are you sure about that? Notice that $x^8 = R(pi/4)^8 = R(2pi) = e$ and the smaller powers are not equal to $e$ so the group should be $$langle xrangle = {e,x,x^2,x^3,x^4,x^5,x^6,x^7}$$
    $endgroup$
    – mechanodroid
    Dec 29 '18 at 12:27










  • $begingroup$
    @Reety: Which book are you using?
    $endgroup$
    – Shaun
    Dec 29 '18 at 12:30










  • $begingroup$
    @Shaun I don't know what it's called since It's a pdf file and it doesn't have the author on it and the names generic.
    $endgroup$
    – Reety
    Dec 29 '18 at 12:36










  • $begingroup$
    @Reety: If you found it online, then copy & paste a long sentence from it into a search engine, with speechmarks both sides; it might show up.
    $endgroup$
    – Shaun
    Dec 29 '18 at 12:40













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


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3055755%2fhow-do-i-work-out-the-generator-for-gl2-bbbr%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









3












$begingroup$

Your matrix $x$ represents a rotation in $mathbb{R}^2$ around the origin by the angle $frac{pi}4$ clockwise.



In general $R(phi) = begin{bmatrix} cosphi & sinphi \ -sinphi & cosphiend{bmatrix}$ does the same thing with the angle $phi$.



Clearly if you rotate by $phi$ and then by $psi$, it is the same as rotating by $phi + psi$. Hence $R(phi)R(psi) = R(phi + psi)$. You can also check this algebraically, of course.



Also $R(phi)^{-1} = R(-phi)$ as the inverse of a rotation by $phi$ is a rotation by $-phi$.



Therefore



$$langle xrangle = {x^n : n in mathbb{Z}} = left{Rleft(frac{pi}4right)^n : n in mathbb{Z}right} = left{Rleft(frac{npi}4right) : n in mathbb{Z}right}$$



Hence your group consists of all rotations by an angle which is a multiple of $fracpi4$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This is a great explanation thank you, although the book I'm reading says the answer is {e, x, x^2, x^3} how can I convert this notation? - @mechanodroid
    $endgroup$
    – Reety
    Dec 29 '18 at 12:16












  • $begingroup$
    @Reety Are you sure about that? Notice that $x^8 = R(pi/4)^8 = R(2pi) = e$ and the smaller powers are not equal to $e$ so the group should be $$langle xrangle = {e,x,x^2,x^3,x^4,x^5,x^6,x^7}$$
    $endgroup$
    – mechanodroid
    Dec 29 '18 at 12:27










  • $begingroup$
    @Reety: Which book are you using?
    $endgroup$
    – Shaun
    Dec 29 '18 at 12:30










  • $begingroup$
    @Shaun I don't know what it's called since It's a pdf file and it doesn't have the author on it and the names generic.
    $endgroup$
    – Reety
    Dec 29 '18 at 12:36










  • $begingroup$
    @Reety: If you found it online, then copy & paste a long sentence from it into a search engine, with speechmarks both sides; it might show up.
    $endgroup$
    – Shaun
    Dec 29 '18 at 12:40


















3












$begingroup$

Your matrix $x$ represents a rotation in $mathbb{R}^2$ around the origin by the angle $frac{pi}4$ clockwise.



In general $R(phi) = begin{bmatrix} cosphi & sinphi \ -sinphi & cosphiend{bmatrix}$ does the same thing with the angle $phi$.



Clearly if you rotate by $phi$ and then by $psi$, it is the same as rotating by $phi + psi$. Hence $R(phi)R(psi) = R(phi + psi)$. You can also check this algebraically, of course.



Also $R(phi)^{-1} = R(-phi)$ as the inverse of a rotation by $phi$ is a rotation by $-phi$.



Therefore



$$langle xrangle = {x^n : n in mathbb{Z}} = left{Rleft(frac{pi}4right)^n : n in mathbb{Z}right} = left{Rleft(frac{npi}4right) : n in mathbb{Z}right}$$



Hence your group consists of all rotations by an angle which is a multiple of $fracpi4$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This is a great explanation thank you, although the book I'm reading says the answer is {e, x, x^2, x^3} how can I convert this notation? - @mechanodroid
    $endgroup$
    – Reety
    Dec 29 '18 at 12:16












  • $begingroup$
    @Reety Are you sure about that? Notice that $x^8 = R(pi/4)^8 = R(2pi) = e$ and the smaller powers are not equal to $e$ so the group should be $$langle xrangle = {e,x,x^2,x^3,x^4,x^5,x^6,x^7}$$
    $endgroup$
    – mechanodroid
    Dec 29 '18 at 12:27










  • $begingroup$
    @Reety: Which book are you using?
    $endgroup$
    – Shaun
    Dec 29 '18 at 12:30










  • $begingroup$
    @Shaun I don't know what it's called since It's a pdf file and it doesn't have the author on it and the names generic.
    $endgroup$
    – Reety
    Dec 29 '18 at 12:36










  • $begingroup$
    @Reety: If you found it online, then copy & paste a long sentence from it into a search engine, with speechmarks both sides; it might show up.
    $endgroup$
    – Shaun
    Dec 29 '18 at 12:40
















3












3








3





$begingroup$

Your matrix $x$ represents a rotation in $mathbb{R}^2$ around the origin by the angle $frac{pi}4$ clockwise.



In general $R(phi) = begin{bmatrix} cosphi & sinphi \ -sinphi & cosphiend{bmatrix}$ does the same thing with the angle $phi$.



Clearly if you rotate by $phi$ and then by $psi$, it is the same as rotating by $phi + psi$. Hence $R(phi)R(psi) = R(phi + psi)$. You can also check this algebraically, of course.



Also $R(phi)^{-1} = R(-phi)$ as the inverse of a rotation by $phi$ is a rotation by $-phi$.



Therefore



$$langle xrangle = {x^n : n in mathbb{Z}} = left{Rleft(frac{pi}4right)^n : n in mathbb{Z}right} = left{Rleft(frac{npi}4right) : n in mathbb{Z}right}$$



Hence your group consists of all rotations by an angle which is a multiple of $fracpi4$.






share|cite|improve this answer









$endgroup$



Your matrix $x$ represents a rotation in $mathbb{R}^2$ around the origin by the angle $frac{pi}4$ clockwise.



In general $R(phi) = begin{bmatrix} cosphi & sinphi \ -sinphi & cosphiend{bmatrix}$ does the same thing with the angle $phi$.



Clearly if you rotate by $phi$ and then by $psi$, it is the same as rotating by $phi + psi$. Hence $R(phi)R(psi) = R(phi + psi)$. You can also check this algebraically, of course.



Also $R(phi)^{-1} = R(-phi)$ as the inverse of a rotation by $phi$ is a rotation by $-phi$.



Therefore



$$langle xrangle = {x^n : n in mathbb{Z}} = left{Rleft(frac{pi}4right)^n : n in mathbb{Z}right} = left{Rleft(frac{npi}4right) : n in mathbb{Z}right}$$



Hence your group consists of all rotations by an angle which is a multiple of $fracpi4$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 29 '18 at 12:10









mechanodroidmechanodroid

27.8k62447




27.8k62447












  • $begingroup$
    This is a great explanation thank you, although the book I'm reading says the answer is {e, x, x^2, x^3} how can I convert this notation? - @mechanodroid
    $endgroup$
    – Reety
    Dec 29 '18 at 12:16












  • $begingroup$
    @Reety Are you sure about that? Notice that $x^8 = R(pi/4)^8 = R(2pi) = e$ and the smaller powers are not equal to $e$ so the group should be $$langle xrangle = {e,x,x^2,x^3,x^4,x^5,x^6,x^7}$$
    $endgroup$
    – mechanodroid
    Dec 29 '18 at 12:27










  • $begingroup$
    @Reety: Which book are you using?
    $endgroup$
    – Shaun
    Dec 29 '18 at 12:30










  • $begingroup$
    @Shaun I don't know what it's called since It's a pdf file and it doesn't have the author on it and the names generic.
    $endgroup$
    – Reety
    Dec 29 '18 at 12:36










  • $begingroup$
    @Reety: If you found it online, then copy & paste a long sentence from it into a search engine, with speechmarks both sides; it might show up.
    $endgroup$
    – Shaun
    Dec 29 '18 at 12:40




















  • $begingroup$
    This is a great explanation thank you, although the book I'm reading says the answer is {e, x, x^2, x^3} how can I convert this notation? - @mechanodroid
    $endgroup$
    – Reety
    Dec 29 '18 at 12:16












  • $begingroup$
    @Reety Are you sure about that? Notice that $x^8 = R(pi/4)^8 = R(2pi) = e$ and the smaller powers are not equal to $e$ so the group should be $$langle xrangle = {e,x,x^2,x^3,x^4,x^5,x^6,x^7}$$
    $endgroup$
    – mechanodroid
    Dec 29 '18 at 12:27










  • $begingroup$
    @Reety: Which book are you using?
    $endgroup$
    – Shaun
    Dec 29 '18 at 12:30










  • $begingroup$
    @Shaun I don't know what it's called since It's a pdf file and it doesn't have the author on it and the names generic.
    $endgroup$
    – Reety
    Dec 29 '18 at 12:36










  • $begingroup$
    @Reety: If you found it online, then copy & paste a long sentence from it into a search engine, with speechmarks both sides; it might show up.
    $endgroup$
    – Shaun
    Dec 29 '18 at 12:40


















$begingroup$
This is a great explanation thank you, although the book I'm reading says the answer is {e, x, x^2, x^3} how can I convert this notation? - @mechanodroid
$endgroup$
– Reety
Dec 29 '18 at 12:16






$begingroup$
This is a great explanation thank you, although the book I'm reading says the answer is {e, x, x^2, x^3} how can I convert this notation? - @mechanodroid
$endgroup$
– Reety
Dec 29 '18 at 12:16














$begingroup$
@Reety Are you sure about that? Notice that $x^8 = R(pi/4)^8 = R(2pi) = e$ and the smaller powers are not equal to $e$ so the group should be $$langle xrangle = {e,x,x^2,x^3,x^4,x^5,x^6,x^7}$$
$endgroup$
– mechanodroid
Dec 29 '18 at 12:27




$begingroup$
@Reety Are you sure about that? Notice that $x^8 = R(pi/4)^8 = R(2pi) = e$ and the smaller powers are not equal to $e$ so the group should be $$langle xrangle = {e,x,x^2,x^3,x^4,x^5,x^6,x^7}$$
$endgroup$
– mechanodroid
Dec 29 '18 at 12:27












$begingroup$
@Reety: Which book are you using?
$endgroup$
– Shaun
Dec 29 '18 at 12:30




$begingroup$
@Reety: Which book are you using?
$endgroup$
– Shaun
Dec 29 '18 at 12:30












$begingroup$
@Shaun I don't know what it's called since It's a pdf file and it doesn't have the author on it and the names generic.
$endgroup$
– Reety
Dec 29 '18 at 12:36




$begingroup$
@Shaun I don't know what it's called since It's a pdf file and it doesn't have the author on it and the names generic.
$endgroup$
– Reety
Dec 29 '18 at 12:36












$begingroup$
@Reety: If you found it online, then copy & paste a long sentence from it into a search engine, with speechmarks both sides; it might show up.
$endgroup$
– Shaun
Dec 29 '18 at 12:40






$begingroup$
@Reety: If you found it online, then copy & paste a long sentence from it into a search engine, with speechmarks both sides; it might show up.
$endgroup$
– Shaun
Dec 29 '18 at 12:40




















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














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3055755%2fhow-do-i-work-out-the-generator-for-gl2-bbbr%23new-answer', 'question_page');
}
);

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







-

Popular posts from this blog

Bressuire

Image
.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid
Read more

Cabo Verde

Image
  Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º)
Read more

Gyllenstierna

Image
.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid
Read more