How is this M(2,2)->R closed under addition?











up vote
1
down vote

favorite












enter image description here



so I understand that It is closed under scalar multiplication, but how is it closed under addition? what if I add another matrix to A? and create new a,b,c,d values?






linear-algebra







share|cite|improve this question


















  • 1




    "Closed under addition" is not quite the right phrase here. We use the properties of being closed under scalar multiplication and closed under addition to define vector spaces (with other properties), but the issue here is whether $T$ is a linear transformation, i.e. a mapping from one vector space to another that preserves operations of scalar multiplication and vector addition.
    – hardmath
    3 hours ago










  • ohhh okay thank you that clarified that for me
    – isuckatprogramming
    2 hours ago















up vote
1
down vote

favorite












enter image description here



so I understand that It is closed under scalar multiplication, but how is it closed under addition? what if I add another matrix to A? and create new a,b,c,d values?






linear-algebra







share|cite|improve this question


















  • 1




    "Closed under addition" is not quite the right phrase here. We use the properties of being closed under scalar multiplication and closed under addition to define vector spaces (with other properties), but the issue here is whether $T$ is a linear transformation, i.e. a mapping from one vector space to another that preserves operations of scalar multiplication and vector addition.
    – hardmath
    3 hours ago










  • ohhh okay thank you that clarified that for me
    – isuckatprogramming
    2 hours ago













up vote
1
down vote

favorite









up vote
1
down vote

favorite











enter image description here



so I understand that It is closed under scalar multiplication, but how is it closed under addition? what if I add another matrix to A? and create new a,b,c,d values?






linear-algebra







share|cite|improve this question













enter image description here



so I understand that It is closed under scalar multiplication, but how is it closed under addition? what if I add another matrix to A? and create new a,b,c,d values?






linear-algebra




linear-algebra






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 3 hours ago









isuckatprogramming

206




206








  • 1




    "Closed under addition" is not quite the right phrase here. We use the properties of being closed under scalar multiplication and closed under addition to define vector spaces (with other properties), but the issue here is whether $T$ is a linear transformation, i.e. a mapping from one vector space to another that preserves operations of scalar multiplication and vector addition.
    – hardmath
    3 hours ago










  • ohhh okay thank you that clarified that for me
    – isuckatprogramming
    2 hours ago














  • 1




    "Closed under addition" is not quite the right phrase here. We use the properties of being closed under scalar multiplication and closed under addition to define vector spaces (with other properties), but the issue here is whether $T$ is a linear transformation, i.e. a mapping from one vector space to another that preserves operations of scalar multiplication and vector addition.
    – hardmath
    3 hours ago










  • ohhh okay thank you that clarified that for me
    – isuckatprogramming
    2 hours ago








1




1




"Closed under addition" is not quite the right phrase here. We use the properties of being closed under scalar multiplication and closed under addition to define vector spaces (with other properties), but the issue here is whether $T$ is a linear transformation, i.e. a mapping from one vector space to another that preserves operations of scalar multiplication and vector addition.
– hardmath
3 hours ago




"Closed under addition" is not quite the right phrase here. We use the properties of being closed under scalar multiplication and closed under addition to define vector spaces (with other properties), but the issue here is whether $T$ is a linear transformation, i.e. a mapping from one vector space to another that preserves operations of scalar multiplication and vector addition.
– hardmath
3 hours ago












ohhh okay thank you that clarified that for me
– isuckatprogramming
2 hours ago




ohhh okay thank you that clarified that for me
– isuckatprogramming
2 hours ago










2 Answers
2






active

oldest

votes

















up vote
5
down vote



accepted










Help me fill this in:



begin{align*}
T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} +T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in M_{2, 2} \
Tleft(begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix}right) &= underline{hspace{50pt}} in mathbb{R}.
end{align*}
Are your answers to questions 3 and 5 the same? If so, then $T$ is additive. You can try a similar tactic with scalar multiplication. Let me know with a comment if you want further help.






share|cite|improve this answer




























    up vote
    0
    down vote













    TIP: the sum of two matrices is a matrix. Try writing $T(A)$ in terms of the values of this matrix and, after expanding each term (because they're bound on the other two matrices), compare the final expression two the other two expressions.






    share|cite|improve this answer





















      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',
      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%2f3044602%2fhow-is-this-m2-2-r-closed-under-addition%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








      up vote
      5
      down vote



      accepted










      Help me fill this in:



      begin{align*}
      T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
      T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
      T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} +T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
      begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in M_{2, 2} \
      Tleft(begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix}right) &= underline{hspace{50pt}} in mathbb{R}.
      end{align*}
      Are your answers to questions 3 and 5 the same? If so, then $T$ is additive. You can try a similar tactic with scalar multiplication. Let me know with a comment if you want further help.






      share|cite|improve this answer

























        up vote
        5
        down vote



        accepted










        Help me fill this in:



        begin{align*}
        T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
        T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
        T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} +T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
        begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in M_{2, 2} \
        Tleft(begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix}right) &= underline{hspace{50pt}} in mathbb{R}.
        end{align*}
        Are your answers to questions 3 and 5 the same? If so, then $T$ is additive. You can try a similar tactic with scalar multiplication. Let me know with a comment if you want further help.






        share|cite|improve this answer























          up vote
          5
          down vote



          accepted







          up vote
          5
          down vote



          accepted






          Help me fill this in:



          begin{align*}
          T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
          T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
          T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} +T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
          begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in M_{2, 2} \
          Tleft(begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix}right) &= underline{hspace{50pt}} in mathbb{R}.
          end{align*}
          Are your answers to questions 3 and 5 the same? If so, then $T$ is additive. You can try a similar tactic with scalar multiplication. Let me know with a comment if you want further help.






          share|cite|improve this answer












          Help me fill this in:



          begin{align*}
          T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
          T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
          T begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} +T begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in mathbb{R} \
          begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix} &= underline{hspace{50pt}} in M_{2, 2} \
          Tleft(begin{pmatrix}a_1 & b_1 \ c_1 & d_1 end{pmatrix} + begin{pmatrix}a_2 & b_2 \ c_2 & d_2 end{pmatrix}right) &= underline{hspace{50pt}} in mathbb{R}.
          end{align*}
          Are your answers to questions 3 and 5 the same? If so, then $T$ is additive. You can try a similar tactic with scalar multiplication. Let me know with a comment if you want further help.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 3 hours ago









          Theo Bendit

          16.2k12148




          16.2k12148






















              up vote
              0
              down vote













              TIP: the sum of two matrices is a matrix. Try writing $T(A)$ in terms of the values of this matrix and, after expanding each term (because they're bound on the other two matrices), compare the final expression two the other two expressions.






              share|cite|improve this answer

























                up vote
                0
                down vote













                TIP: the sum of two matrices is a matrix. Try writing $T(A)$ in terms of the values of this matrix and, after expanding each term (because they're bound on the other two matrices), compare the final expression two the other two expressions.






                share|cite|improve this answer























                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  TIP: the sum of two matrices is a matrix. Try writing $T(A)$ in terms of the values of this matrix and, after expanding each term (because they're bound on the other two matrices), compare the final expression two the other two expressions.






                  share|cite|improve this answer












                  TIP: the sum of two matrices is a matrix. Try writing $T(A)$ in terms of the values of this matrix and, after expanding each term (because they're bound on the other two matrices), compare the final expression two the other two expressions.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 3 hours ago









                  Lucas Henrique

                  723312




                  723312






























                      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.





                      Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                      Please pay close attention to the following guidance:


                      • 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.


                      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%2f3044602%2fhow-is-this-m2-2-r-closed-under-addition%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

                      Costa Masnaga

                      Image
                      .mw-parser-output .avviso .mbox-text-div>div,.mw-parser-output .avviso .mbox-text-full-div>div{font-size:90%}.mw-parser-output .avviso .mbox-image div{width:52px}.mw-parser-output .avviso .mbox-text-full-div .hide-when-compact{display:block} Questa voce o sezione sull'argomento Lombardia non cita le fonti necessarie o quelle presenti sono insufficienti . Puoi migliorare questa voce aggiungendo citazioni da fonti attendibili secondo le linee guida sull'uso delle fonti. Segui i suggerimenti del progetto di riferimento. Costa Masnaga comune Localizzazione Stato   Italia Regione Lombardia Provincia Lecco Amministrazione Sindaco Sabina Panzeri (Al centro il cittadino) dall'08/06/2009 Territorio Coordinate 45°46′N 9°17′E  /  45.766667°N 9.283333°E 45.766667; 9.283333  ( Costa Masnaga ) Coordinate: 45°46′N 9°17′E  /  45.766667°N 9.283333°E 45.766667; 9.283333  ( Costa Masnaga ) Altitudine 318 m s.l.m...
                      Read more

                      Fotorealismo

                      Image
                      Guerrino Boatto, Venice's St. Barnaba Church and canal Il fotorealismo è un genere di pittura basato sull'uso di una o più fotografie dalle quali l'artista prende le informazioni necessarie da utilizzare poi nel processo della creazione dell'opera, in genere ad olio o ad acrilico, al fine di avere un aspetto finale il più simile possibile alla fotografia. Indice 1 Storia 2 Lista dei fotorealisti 3 Note 4 Voci correlate 5 Altri progetti Storia | Il fotorealismo è un movimento che nasce negli Stati Uniti d'America alla fine degli anni sessanta [1] sulla scia della Pop Art [2] [3] [4] , e insieme condividono il carattere reazionario verso la travolgente ascesa dei media, che alla metà del ventesimo secolo si trasforma in un fenomeno di massa talmente imponente da minacciare il valore dell'immagine nell'arte. [5] [6] Ne consegue così che, colto il gesto d'inizio che fu della Pop Art, i fotorealisti aprono ad un ...
                      Read more

                      Sidney Franklin

                      Image
                      Questa voce sull'argomento registi statunitensi è solo un abbozzo . Contribuisci a migliorarla secondo le convenzioni di Wikipedia. Sidney Franklin, foto di Carl Van Vechten Oscar alla memoria Irving G. Thalberg 1943 Sidney Franklin , all'anagrafe Sidney Arnold Franklin , (San Francisco, 21 marzo 1893 – Santa Monica, 18 maggio 1972), è stato un regista, produttore cinematografico e attore statunitense. Il 20 giugno 1963, si sposò con l'attrice australiana Enid Bennett. Il matrimonio durò fino alla morte dell'attrice, il 14 maggio 1969 [1] Indice 1 Filmografia 1.1 Regista 1.2 Aiuto regia (parziale) 1.3 Produttore 1.4 Attore (parziale) 2 Note 3 Altri progetti 4 Collegamenti esterni Filmografia | Regista | The Baby , co-regia Chester M. Franklin (1915) The Rivals , co-regia Chester M. Franklin (1915) Little Dick's First Case , co-regia Chester M. Franklin (1915) Her Filmland Hero , co-...
                      Read more