Linear Transformation involving polynomial and matrix












2












$begingroup$


Determine if its linear the transformation



$f:Re_{3} [x]rightarrow M_{2x2}$



such as



$f(ax^3+bx^2+cx+d) = begin{bmatrix}
a-c & 0 \
0&b+d
end{bmatrix}$



for any $ax^3+bx^2+cx+d in Re _{3} [x]$



Im having trouble associating the polynomial with the resulting matrix.
I know I'm supposed to check the two main Linear transformation conditions but I don't know how to aproach this example.



Here's my attempt:



Condition 1: $T(u+v)=T(u)+T(v)$



$f((ax^3+bx^2+cx+d)+(a'x^3+b'x^2+c'x+d'))=f((ax^3+a'x^3)+(bx^2+b'x^2)+(cx+c'x)+(d+d'))=f((a+a')x^3+(b+b')x^2+(c+c')x+(d+d'))=begin{bmatrix}
(a+a')-(c+c') & 0\
0 & (b+b')+(d+d')
end{bmatrix} = begin{bmatrix}
a-c & 0\
0 & b+d
end{bmatrix} + begin{bmatrix}
a'-c' & 0\
0 & b'+d'
end{bmatrix}=f(ax^3+bx^2+cx+d)+f(a'x^3+b'x^2+c'x+d')$



Is this correct?






linear-transformations







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    Looks right to me.
    $endgroup$
    – saulspatz
    Jan 5 at 18:33






  • 3




    $begingroup$
    Seems fine. ${}{}{}{} $
    $endgroup$
    – Thomas Shelby
    Jan 5 at 18:33
















2












$begingroup$


Determine if its linear the transformation



$f:Re_{3} [x]rightarrow M_{2x2}$



such as



$f(ax^3+bx^2+cx+d) = begin{bmatrix}
a-c & 0 \
0&b+d
end{bmatrix}$



for any $ax^3+bx^2+cx+d in Re _{3} [x]$



Im having trouble associating the polynomial with the resulting matrix.
I know I'm supposed to check the two main Linear transformation conditions but I don't know how to aproach this example.



Here's my attempt:



Condition 1: $T(u+v)=T(u)+T(v)$



$f((ax^3+bx^2+cx+d)+(a'x^3+b'x^2+c'x+d'))=f((ax^3+a'x^3)+(bx^2+b'x^2)+(cx+c'x)+(d+d'))=f((a+a')x^3+(b+b')x^2+(c+c')x+(d+d'))=begin{bmatrix}
(a+a')-(c+c') & 0\
0 & (b+b')+(d+d')
end{bmatrix} = begin{bmatrix}
a-c & 0\
0 & b+d
end{bmatrix} + begin{bmatrix}
a'-c' & 0\
0 & b'+d'
end{bmatrix}=f(ax^3+bx^2+cx+d)+f(a'x^3+b'x^2+c'x+d')$



Is this correct?






linear-transformations







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    Looks right to me.
    $endgroup$
    – saulspatz
    Jan 5 at 18:33






  • 3




    $begingroup$
    Seems fine. ${}{}{}{} $
    $endgroup$
    – Thomas Shelby
    Jan 5 at 18:33














2












2








2





$begingroup$


Determine if its linear the transformation



$f:Re_{3} [x]rightarrow M_{2x2}$



such as



$f(ax^3+bx^2+cx+d) = begin{bmatrix}
a-c & 0 \
0&b+d
end{bmatrix}$



for any $ax^3+bx^2+cx+d in Re _{3} [x]$



Im having trouble associating the polynomial with the resulting matrix.
I know I'm supposed to check the two main Linear transformation conditions but I don't know how to aproach this example.



Here's my attempt:



Condition 1: $T(u+v)=T(u)+T(v)$



$f((ax^3+bx^2+cx+d)+(a'x^3+b'x^2+c'x+d'))=f((ax^3+a'x^3)+(bx^2+b'x^2)+(cx+c'x)+(d+d'))=f((a+a')x^3+(b+b')x^2+(c+c')x+(d+d'))=begin{bmatrix}
(a+a')-(c+c') & 0\
0 & (b+b')+(d+d')
end{bmatrix} = begin{bmatrix}
a-c & 0\
0 & b+d
end{bmatrix} + begin{bmatrix}
a'-c' & 0\
0 & b'+d'
end{bmatrix}=f(ax^3+bx^2+cx+d)+f(a'x^3+b'x^2+c'x+d')$



Is this correct?






linear-transformations







share|cite|improve this question









$endgroup$




Determine if its linear the transformation



$f:Re_{3} [x]rightarrow M_{2x2}$



such as



$f(ax^3+bx^2+cx+d) = begin{bmatrix}
a-c & 0 \
0&b+d
end{bmatrix}$



for any $ax^3+bx^2+cx+d in Re _{3} [x]$



Im having trouble associating the polynomial with the resulting matrix.
I know I'm supposed to check the two main Linear transformation conditions but I don't know how to aproach this example.



Here's my attempt:



Condition 1: $T(u+v)=T(u)+T(v)$



$f((ax^3+bx^2+cx+d)+(a'x^3+b'x^2+c'x+d'))=f((ax^3+a'x^3)+(bx^2+b'x^2)+(cx+c'x)+(d+d'))=f((a+a')x^3+(b+b')x^2+(c+c')x+(d+d'))=begin{bmatrix}
(a+a')-(c+c') & 0\
0 & (b+b')+(d+d')
end{bmatrix} = begin{bmatrix}
a-c & 0\
0 & b+d
end{bmatrix} + begin{bmatrix}
a'-c' & 0\
0 & b'+d'
end{bmatrix}=f(ax^3+bx^2+cx+d)+f(a'x^3+b'x^2+c'x+d')$



Is this correct?






linear-transformations




linear-transformations






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 5 at 18:29









JakcjonesJakcjones

828




828








  • 3




    $begingroup$
    Looks right to me.
    $endgroup$
    – saulspatz
    Jan 5 at 18:33






  • 3




    $begingroup$
    Seems fine. ${}{}{}{} $
    $endgroup$
    – Thomas Shelby
    Jan 5 at 18:33














  • 3




    $begingroup$
    Looks right to me.
    $endgroup$
    – saulspatz
    Jan 5 at 18:33






  • 3




    $begingroup$
    Seems fine. ${}{}{}{} $
    $endgroup$
    – Thomas Shelby
    Jan 5 at 18:33








3




3




$begingroup$
Looks right to me.
$endgroup$
– saulspatz
Jan 5 at 18:33




$begingroup$
Looks right to me.
$endgroup$
– saulspatz
Jan 5 at 18:33




3




3




$begingroup$
Seems fine. ${}{}{}{} $
$endgroup$
– Thomas Shelby
Jan 5 at 18:33




$begingroup$
Seems fine. ${}{}{}{} $
$endgroup$
– Thomas Shelby
Jan 5 at 18:33










1 Answer
1






active

oldest

votes


















2












$begingroup$

That looks fine. Another way to go is this:



Consider the standard basis $$bigl{A_{1,1},A_{2,1},A_{1,2},A_{2,2}bigr}$$ for $M_{2times 2},$ where $A_{i,j}$ is the $2times 2$ matrix of zeroes, except the $i$th row $j$th column entry, which is $1.$ Considering also the standard basis ${x^3,x^2,x,1}$ for $mathfrak{R}_3[x],$ then the given transformation has the matrix representation $$begin{bmatrix}1 & 0 & -1 & 0\0 & 0 & 0 & 0\0 & 0 & 0 & 0\0 & 1 & 0 & 1end{bmatrix}.$$



Since it has a matrix representation, then it's a linear transformation.






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%2f3063041%2flinear-transformation-involving-polynomial-and-matrix%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









    2












    $begingroup$

    That looks fine. Another way to go is this:



    Consider the standard basis $$bigl{A_{1,1},A_{2,1},A_{1,2},A_{2,2}bigr}$$ for $M_{2times 2},$ where $A_{i,j}$ is the $2times 2$ matrix of zeroes, except the $i$th row $j$th column entry, which is $1.$ Considering also the standard basis ${x^3,x^2,x,1}$ for $mathfrak{R}_3[x],$ then the given transformation has the matrix representation $$begin{bmatrix}1 & 0 & -1 & 0\0 & 0 & 0 & 0\0 & 0 & 0 & 0\0 & 1 & 0 & 1end{bmatrix}.$$



    Since it has a matrix representation, then it's a linear transformation.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      That looks fine. Another way to go is this:



      Consider the standard basis $$bigl{A_{1,1},A_{2,1},A_{1,2},A_{2,2}bigr}$$ for $M_{2times 2},$ where $A_{i,j}$ is the $2times 2$ matrix of zeroes, except the $i$th row $j$th column entry, which is $1.$ Considering also the standard basis ${x^3,x^2,x,1}$ for $mathfrak{R}_3[x],$ then the given transformation has the matrix representation $$begin{bmatrix}1 & 0 & -1 & 0\0 & 0 & 0 & 0\0 & 0 & 0 & 0\0 & 1 & 0 & 1end{bmatrix}.$$



      Since it has a matrix representation, then it's a linear transformation.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        That looks fine. Another way to go is this:



        Consider the standard basis $$bigl{A_{1,1},A_{2,1},A_{1,2},A_{2,2}bigr}$$ for $M_{2times 2},$ where $A_{i,j}$ is the $2times 2$ matrix of zeroes, except the $i$th row $j$th column entry, which is $1.$ Considering also the standard basis ${x^3,x^2,x,1}$ for $mathfrak{R}_3[x],$ then the given transformation has the matrix representation $$begin{bmatrix}1 & 0 & -1 & 0\0 & 0 & 0 & 0\0 & 0 & 0 & 0\0 & 1 & 0 & 1end{bmatrix}.$$



        Since it has a matrix representation, then it's a linear transformation.






        share|cite|improve this answer









        $endgroup$



        That looks fine. Another way to go is this:



        Consider the standard basis $$bigl{A_{1,1},A_{2,1},A_{1,2},A_{2,2}bigr}$$ for $M_{2times 2},$ where $A_{i,j}$ is the $2times 2$ matrix of zeroes, except the $i$th row $j$th column entry, which is $1.$ Considering also the standard basis ${x^3,x^2,x,1}$ for $mathfrak{R}_3[x],$ then the given transformation has the matrix representation $$begin{bmatrix}1 & 0 & -1 & 0\0 & 0 & 0 & 0\0 & 0 & 0 & 0\0 & 1 & 0 & 1end{bmatrix}.$$



        Since it has a matrix representation, then it's a linear transformation.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 5 at 19:30









        Cameron BuieCameron Buie

        85.9k772161




        85.9k772161






























            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%2f3063041%2flinear-transformation-involving-polynomial-and-matrix%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