Functions over a $C$ vector space with geometric importance. (How to find the basis?)












1












$begingroup$


Searching through our suggested exercises of linear and abstract algebra for solving, I found the following exercise. The reason I am posting this, is that because we haven't went through complex spaces and theory but the exercise is still in the suggested ones, which means I am missing something :



Let $V$ be the $C$-vector space of the complex functions over the set of the vertexes of a square. We define a linear operator $T$ over $V$ to be :



$Tf(x) = f(r(x)) $



where for a vertex $x$, $r(x)$ is the exactly next vertex (going clockwise).



->Describe a basis of $V$, write the matrix of the operator $T$ over the particular base you found, find the eigenvalues and describe the eigenvectors of $T$ as functions over the vertex of the square.



What I searched and tried was :



First of all, going clockwise and since we are speaking for a complex vector space, from what I know I think that $r(x) = -ix$ (correct me if that's wrong).



Also, $f$ is a functional defined as : $f : mathbb C^m -> mathbb C$ and $T$ is an operator defined as : $T: V->mathbb C$. From the definition of the operator and the functional (if I did it correctly), how will I procced to finding the basis? I believe that once I define the basis, I will not have an issue solving the rest of the exercise because it goes as a chain, since you need the basis to find the matrix, then you need the matrix to find the eigenvalues and the eigenvalues for the eigenvectors. Given that I know perfectly the theory for bases, matrices, eigenvalues and eigenvectors for the space $mathbb R$ of the real numbers, I think that by scrolling through complex-algebra parts of the books I won't have a problem solving it. But I couldn't find such a type of exercise so I can understand how to proceed with the basis in the $mathbb C$-complex space. I would really appreciate if you could explain how to find the basis, the rest I will finish them, but that's where I am stuck at (the basis).






linear-algebra abstract-algebra functional-analysis operator-theory complex-geometry







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    A square has four vertices. Your space $V$ contains functions that assign a (complex) number to each of these four vertices. So an element of $V$ is just a tuple of four numbers! Does this sound familiar? Now can you give me a basis for $V$? Don't let the complex part bother you too much. It will not affect anything until the finding of the eigenvalues.
    $endgroup$
    – Ravi
    May 3 '16 at 19:58












  • $begingroup$
    Oh, let me give it a hand that way then !
    $endgroup$
    – Rebellos
    May 3 '16 at 20:10










  • $begingroup$
    By the way, what does "tuple" mean ? Because even the greek dictionary word is something that we don't use anymore :P
    $endgroup$
    – Rebellos
    May 3 '16 at 20:11










  • $begingroup$
    An $n$-tuple is a list of $n$ numbers. I.e. $(a_1,cdots ,a_n)$.
    $endgroup$
    – Ravi
    May 3 '16 at 20:13












  • $begingroup$
    Oh I get it now. So we have a vector over that space which is $(a_1,a_2,a_3,a_4) $ . Now, the complex number which is assigned, is assigned by the way I thought it did ? $ r(x) = -ix$ ? Then going on like : $Tf(x) = f(-ix) $
    $endgroup$
    – Rebellos
    May 3 '16 at 20:17


















1












$begingroup$


Searching through our suggested exercises of linear and abstract algebra for solving, I found the following exercise. The reason I am posting this, is that because we haven't went through complex spaces and theory but the exercise is still in the suggested ones, which means I am missing something :



Let $V$ be the $C$-vector space of the complex functions over the set of the vertexes of a square. We define a linear operator $T$ over $V$ to be :



$Tf(x) = f(r(x)) $



where for a vertex $x$, $r(x)$ is the exactly next vertex (going clockwise).



->Describe a basis of $V$, write the matrix of the operator $T$ over the particular base you found, find the eigenvalues and describe the eigenvectors of $T$ as functions over the vertex of the square.



What I searched and tried was :



First of all, going clockwise and since we are speaking for a complex vector space, from what I know I think that $r(x) = -ix$ (correct me if that's wrong).



Also, $f$ is a functional defined as : $f : mathbb C^m -> mathbb C$ and $T$ is an operator defined as : $T: V->mathbb C$. From the definition of the operator and the functional (if I did it correctly), how will I procced to finding the basis? I believe that once I define the basis, I will not have an issue solving the rest of the exercise because it goes as a chain, since you need the basis to find the matrix, then you need the matrix to find the eigenvalues and the eigenvalues for the eigenvectors. Given that I know perfectly the theory for bases, matrices, eigenvalues and eigenvectors for the space $mathbb R$ of the real numbers, I think that by scrolling through complex-algebra parts of the books I won't have a problem solving it. But I couldn't find such a type of exercise so I can understand how to proceed with the basis in the $mathbb C$-complex space. I would really appreciate if you could explain how to find the basis, the rest I will finish them, but that's where I am stuck at (the basis).






linear-algebra abstract-algebra functional-analysis operator-theory complex-geometry







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    A square has four vertices. Your space $V$ contains functions that assign a (complex) number to each of these four vertices. So an element of $V$ is just a tuple of four numbers! Does this sound familiar? Now can you give me a basis for $V$? Don't let the complex part bother you too much. It will not affect anything until the finding of the eigenvalues.
    $endgroup$
    – Ravi
    May 3 '16 at 19:58












  • $begingroup$
    Oh, let me give it a hand that way then !
    $endgroup$
    – Rebellos
    May 3 '16 at 20:10










  • $begingroup$
    By the way, what does "tuple" mean ? Because even the greek dictionary word is something that we don't use anymore :P
    $endgroup$
    – Rebellos
    May 3 '16 at 20:11










  • $begingroup$
    An $n$-tuple is a list of $n$ numbers. I.e. $(a_1,cdots ,a_n)$.
    $endgroup$
    – Ravi
    May 3 '16 at 20:13












  • $begingroup$
    Oh I get it now. So we have a vector over that space which is $(a_1,a_2,a_3,a_4) $ . Now, the complex number which is assigned, is assigned by the way I thought it did ? $ r(x) = -ix$ ? Then going on like : $Tf(x) = f(-ix) $
    $endgroup$
    – Rebellos
    May 3 '16 at 20:17
















1












1








1





$begingroup$


Searching through our suggested exercises of linear and abstract algebra for solving, I found the following exercise. The reason I am posting this, is that because we haven't went through complex spaces and theory but the exercise is still in the suggested ones, which means I am missing something :



Let $V$ be the $C$-vector space of the complex functions over the set of the vertexes of a square. We define a linear operator $T$ over $V$ to be :



$Tf(x) = f(r(x)) $



where for a vertex $x$, $r(x)$ is the exactly next vertex (going clockwise).



->Describe a basis of $V$, write the matrix of the operator $T$ over the particular base you found, find the eigenvalues and describe the eigenvectors of $T$ as functions over the vertex of the square.



What I searched and tried was :



First of all, going clockwise and since we are speaking for a complex vector space, from what I know I think that $r(x) = -ix$ (correct me if that's wrong).



Also, $f$ is a functional defined as : $f : mathbb C^m -> mathbb C$ and $T$ is an operator defined as : $T: V->mathbb C$. From the definition of the operator and the functional (if I did it correctly), how will I procced to finding the basis? I believe that once I define the basis, I will not have an issue solving the rest of the exercise because it goes as a chain, since you need the basis to find the matrix, then you need the matrix to find the eigenvalues and the eigenvalues for the eigenvectors. Given that I know perfectly the theory for bases, matrices, eigenvalues and eigenvectors for the space $mathbb R$ of the real numbers, I think that by scrolling through complex-algebra parts of the books I won't have a problem solving it. But I couldn't find such a type of exercise so I can understand how to proceed with the basis in the $mathbb C$-complex space. I would really appreciate if you could explain how to find the basis, the rest I will finish them, but that's where I am stuck at (the basis).






linear-algebra abstract-algebra functional-analysis operator-theory complex-geometry







share|cite|improve this question









$endgroup$




Searching through our suggested exercises of linear and abstract algebra for solving, I found the following exercise. The reason I am posting this, is that because we haven't went through complex spaces and theory but the exercise is still in the suggested ones, which means I am missing something :



Let $V$ be the $C$-vector space of the complex functions over the set of the vertexes of a square. We define a linear operator $T$ over $V$ to be :



$Tf(x) = f(r(x)) $



where for a vertex $x$, $r(x)$ is the exactly next vertex (going clockwise).



->Describe a basis of $V$, write the matrix of the operator $T$ over the particular base you found, find the eigenvalues and describe the eigenvectors of $T$ as functions over the vertex of the square.



What I searched and tried was :



First of all, going clockwise and since we are speaking for a complex vector space, from what I know I think that $r(x) = -ix$ (correct me if that's wrong).



Also, $f$ is a functional defined as : $f : mathbb C^m -> mathbb C$ and $T$ is an operator defined as : $T: V->mathbb C$. From the definition of the operator and the functional (if I did it correctly), how will I procced to finding the basis? I believe that once I define the basis, I will not have an issue solving the rest of the exercise because it goes as a chain, since you need the basis to find the matrix, then you need the matrix to find the eigenvalues and the eigenvalues for the eigenvectors. Given that I know perfectly the theory for bases, matrices, eigenvalues and eigenvectors for the space $mathbb R$ of the real numbers, I think that by scrolling through complex-algebra parts of the books I won't have a problem solving it. But I couldn't find such a type of exercise so I can understand how to proceed with the basis in the $mathbb C$-complex space. I would really appreciate if you could explain how to find the basis, the rest I will finish them, but that's where I am stuck at (the basis).






linear-algebra abstract-algebra functional-analysis operator-theory complex-geometry




linear-algebra abstract-algebra functional-analysis operator-theory complex-geometry






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked May 3 '16 at 19:53









RebellosRebellos

14.5k31246




14.5k31246








  • 1




    $begingroup$
    A square has four vertices. Your space $V$ contains functions that assign a (complex) number to each of these four vertices. So an element of $V$ is just a tuple of four numbers! Does this sound familiar? Now can you give me a basis for $V$? Don't let the complex part bother you too much. It will not affect anything until the finding of the eigenvalues.
    $endgroup$
    – Ravi
    May 3 '16 at 19:58












  • $begingroup$
    Oh, let me give it a hand that way then !
    $endgroup$
    – Rebellos
    May 3 '16 at 20:10










  • $begingroup$
    By the way, what does "tuple" mean ? Because even the greek dictionary word is something that we don't use anymore :P
    $endgroup$
    – Rebellos
    May 3 '16 at 20:11










  • $begingroup$
    An $n$-tuple is a list of $n$ numbers. I.e. $(a_1,cdots ,a_n)$.
    $endgroup$
    – Ravi
    May 3 '16 at 20:13












  • $begingroup$
    Oh I get it now. So we have a vector over that space which is $(a_1,a_2,a_3,a_4) $ . Now, the complex number which is assigned, is assigned by the way I thought it did ? $ r(x) = -ix$ ? Then going on like : $Tf(x) = f(-ix) $
    $endgroup$
    – Rebellos
    May 3 '16 at 20:17
















  • 1




    $begingroup$
    A square has four vertices. Your space $V$ contains functions that assign a (complex) number to each of these four vertices. So an element of $V$ is just a tuple of four numbers! Does this sound familiar? Now can you give me a basis for $V$? Don't let the complex part bother you too much. It will not affect anything until the finding of the eigenvalues.
    $endgroup$
    – Ravi
    May 3 '16 at 19:58












  • $begingroup$
    Oh, let me give it a hand that way then !
    $endgroup$
    – Rebellos
    May 3 '16 at 20:10










  • $begingroup$
    By the way, what does "tuple" mean ? Because even the greek dictionary word is something that we don't use anymore :P
    $endgroup$
    – Rebellos
    May 3 '16 at 20:11










  • $begingroup$
    An $n$-tuple is a list of $n$ numbers. I.e. $(a_1,cdots ,a_n)$.
    $endgroup$
    – Ravi
    May 3 '16 at 20:13












  • $begingroup$
    Oh I get it now. So we have a vector over that space which is $(a_1,a_2,a_3,a_4) $ . Now, the complex number which is assigned, is assigned by the way I thought it did ? $ r(x) = -ix$ ? Then going on like : $Tf(x) = f(-ix) $
    $endgroup$
    – Rebellos
    May 3 '16 at 20:17










1




1




$begingroup$
A square has four vertices. Your space $V$ contains functions that assign a (complex) number to each of these four vertices. So an element of $V$ is just a tuple of four numbers! Does this sound familiar? Now can you give me a basis for $V$? Don't let the complex part bother you too much. It will not affect anything until the finding of the eigenvalues.
$endgroup$
– Ravi
May 3 '16 at 19:58






$begingroup$
A square has four vertices. Your space $V$ contains functions that assign a (complex) number to each of these four vertices. So an element of $V$ is just a tuple of four numbers! Does this sound familiar? Now can you give me a basis for $V$? Don't let the complex part bother you too much. It will not affect anything until the finding of the eigenvalues.
$endgroup$
– Ravi
May 3 '16 at 19:58














$begingroup$
Oh, let me give it a hand that way then !
$endgroup$
– Rebellos
May 3 '16 at 20:10




$begingroup$
Oh, let me give it a hand that way then !
$endgroup$
– Rebellos
May 3 '16 at 20:10












$begingroup$
By the way, what does "tuple" mean ? Because even the greek dictionary word is something that we don't use anymore :P
$endgroup$
– Rebellos
May 3 '16 at 20:11




$begingroup$
By the way, what does "tuple" mean ? Because even the greek dictionary word is something that we don't use anymore :P
$endgroup$
– Rebellos
May 3 '16 at 20:11












$begingroup$
An $n$-tuple is a list of $n$ numbers. I.e. $(a_1,cdots ,a_n)$.
$endgroup$
– Ravi
May 3 '16 at 20:13






$begingroup$
An $n$-tuple is a list of $n$ numbers. I.e. $(a_1,cdots ,a_n)$.
$endgroup$
– Ravi
May 3 '16 at 20:13














$begingroup$
Oh I get it now. So we have a vector over that space which is $(a_1,a_2,a_3,a_4) $ . Now, the complex number which is assigned, is assigned by the way I thought it did ? $ r(x) = -ix$ ? Then going on like : $Tf(x) = f(-ix) $
$endgroup$
– Rebellos
May 3 '16 at 20:17






$begingroup$
Oh I get it now. So we have a vector over that space which is $(a_1,a_2,a_3,a_4) $ . Now, the complex number which is assigned, is assigned by the way I thought it did ? $ r(x) = -ix$ ? Then going on like : $Tf(x) = f(-ix) $
$endgroup$
– Rebellos
May 3 '16 at 20:17












1 Answer
1






active

oldest

votes


















1












$begingroup$

Answering to my question after a great help from Jake, through comments and chat :



Let's say that $(a_1,a_2,a_3,a_4)$ are the four vertices of our square. Then, the operator that assigns them exactly to the next one will give back the vector $(a_2,a_3,a_4,a_1)$. (Note that the fourth one goes back to the first one). Then, a vector $v$ of the space $mathbb C^4$ that our vertices are assigned, is written as : $v = e_1a_2 + e_2a_3 + e_3a_4 + e_4a_1$.



From that, we get that the basis $B$ is the : $B=text{span}[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]$



And furthermore, the matrix of our operator T :



$ M_T =
begin{bmatrix}
0 & 0 & 0 & 1 \
0 & 0 & 1 & 0 \
0 & 1 & 0 & 0 \
1 & 0 & 0 & 0
end{bmatrix} $



We get that exact matrix, because $(1,0,0,0)$ goes to $(0,0,0,1)$ etc.
After that, we can move on to solving the characteristic polynomial and finding the eigenvalues and eigenvectors.






share|cite|improve this answer











$endgroup$













    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%2f1770293%2ffunctions-over-a-c-vector-space-with-geometric-importance-how-to-find-the-ba%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












    $begingroup$

    Answering to my question after a great help from Jake, through comments and chat :



    Let's say that $(a_1,a_2,a_3,a_4)$ are the four vertices of our square. Then, the operator that assigns them exactly to the next one will give back the vector $(a_2,a_3,a_4,a_1)$. (Note that the fourth one goes back to the first one). Then, a vector $v$ of the space $mathbb C^4$ that our vertices are assigned, is written as : $v = e_1a_2 + e_2a_3 + e_3a_4 + e_4a_1$.



    From that, we get that the basis $B$ is the : $B=text{span}[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]$



    And furthermore, the matrix of our operator T :



    $ M_T =
    begin{bmatrix}
    0 & 0 & 0 & 1 \
    0 & 0 & 1 & 0 \
    0 & 1 & 0 & 0 \
    1 & 0 & 0 & 0
    end{bmatrix} $



    We get that exact matrix, because $(1,0,0,0)$ goes to $(0,0,0,1)$ etc.
    After that, we can move on to solving the characteristic polynomial and finding the eigenvalues and eigenvectors.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      Answering to my question after a great help from Jake, through comments and chat :



      Let's say that $(a_1,a_2,a_3,a_4)$ are the four vertices of our square. Then, the operator that assigns them exactly to the next one will give back the vector $(a_2,a_3,a_4,a_1)$. (Note that the fourth one goes back to the first one). Then, a vector $v$ of the space $mathbb C^4$ that our vertices are assigned, is written as : $v = e_1a_2 + e_2a_3 + e_3a_4 + e_4a_1$.



      From that, we get that the basis $B$ is the : $B=text{span}[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]$



      And furthermore, the matrix of our operator T :



      $ M_T =
      begin{bmatrix}
      0 & 0 & 0 & 1 \
      0 & 0 & 1 & 0 \
      0 & 1 & 0 & 0 \
      1 & 0 & 0 & 0
      end{bmatrix} $



      We get that exact matrix, because $(1,0,0,0)$ goes to $(0,0,0,1)$ etc.
      After that, we can move on to solving the characteristic polynomial and finding the eigenvalues and eigenvectors.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        Answering to my question after a great help from Jake, through comments and chat :



        Let's say that $(a_1,a_2,a_3,a_4)$ are the four vertices of our square. Then, the operator that assigns them exactly to the next one will give back the vector $(a_2,a_3,a_4,a_1)$. (Note that the fourth one goes back to the first one). Then, a vector $v$ of the space $mathbb C^4$ that our vertices are assigned, is written as : $v = e_1a_2 + e_2a_3 + e_3a_4 + e_4a_1$.



        From that, we get that the basis $B$ is the : $B=text{span}[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]$



        And furthermore, the matrix of our operator T :



        $ M_T =
        begin{bmatrix}
        0 & 0 & 0 & 1 \
        0 & 0 & 1 & 0 \
        0 & 1 & 0 & 0 \
        1 & 0 & 0 & 0
        end{bmatrix} $



        We get that exact matrix, because $(1,0,0,0)$ goes to $(0,0,0,1)$ etc.
        After that, we can move on to solving the characteristic polynomial and finding the eigenvalues and eigenvectors.






        share|cite|improve this answer











        $endgroup$



        Answering to my question after a great help from Jake, through comments and chat :



        Let's say that $(a_1,a_2,a_3,a_4)$ are the four vertices of our square. Then, the operator that assigns them exactly to the next one will give back the vector $(a_2,a_3,a_4,a_1)$. (Note that the fourth one goes back to the first one). Then, a vector $v$ of the space $mathbb C^4$ that our vertices are assigned, is written as : $v = e_1a_2 + e_2a_3 + e_3a_4 + e_4a_1$.



        From that, we get that the basis $B$ is the : $B=text{span}[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]$



        And furthermore, the matrix of our operator T :



        $ M_T =
        begin{bmatrix}
        0 & 0 & 0 & 1 \
        0 & 0 & 1 & 0 \
        0 & 1 & 0 & 0 \
        1 & 0 & 0 & 0
        end{bmatrix} $



        We get that exact matrix, because $(1,0,0,0)$ goes to $(0,0,0,1)$ etc.
        After that, we can move on to solving the characteristic polynomial and finding the eigenvalues and eigenvectors.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 16 '18 at 19:53

























        answered May 3 '16 at 21:36









        RebellosRebellos

        14.5k31246




        14.5k31246






























            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%2f1770293%2ffunctions-over-a-c-vector-space-with-geometric-importance-how-to-find-the-ba%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