What is the difference between a zero operator, zero function, zero scalar, and zero vector?












2












$begingroup$


I'm pretty sure that a zero vector is just a vector of length zero with direction, zero scalar is just the number zero, and that a zero function is any function that maps to zero. Not entirely sure what exactly a zero operator is however.






linear-algebra soft-question terminology







share|cite|improve this question









New contributor




Arlene is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$

















    2












    $begingroup$


    I'm pretty sure that a zero vector is just a vector of length zero with direction, zero scalar is just the number zero, and that a zero function is any function that maps to zero. Not entirely sure what exactly a zero operator is however.






    linear-algebra soft-question terminology







    share|cite|improve this question









    New contributor




    Arlene is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.







    $endgroup$















      2












      2








      2


      1



      $begingroup$


      I'm pretty sure that a zero vector is just a vector of length zero with direction, zero scalar is just the number zero, and that a zero function is any function that maps to zero. Not entirely sure what exactly a zero operator is however.






      linear-algebra soft-question terminology







      share|cite|improve this question









      New contributor




      Arlene is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      I'm pretty sure that a zero vector is just a vector of length zero with direction, zero scalar is just the number zero, and that a zero function is any function that maps to zero. Not entirely sure what exactly a zero operator is however.






      linear-algebra soft-question terminology




      linear-algebra soft-question terminology






      share|cite|improve this question









      New contributor




      Arlene is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question









      New contributor




      Arlene is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question








      edited 4 hours ago









      J. W. Tanner

      2,0691117




      2,0691117






      New contributor




      Arlene is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      asked 4 hours ago









      ArleneArlene

      133




      133




      New contributor




      Arlene is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.





      New contributor





      Arlene is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






      Arlene is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






















          2 Answers
          2






          active

          oldest

          votes


















          4












          $begingroup$

          The zero vector is a vector, i.e. a member of whatever vector space is under consideration. It has the property that adding it to any vector $bf v$ in the vector space leaves $bf v$ unchanged.



          The zero scalar is a scalar, i.e. a member of the field that is part of the definition of the vector space (usually the real or complex numbers in an elementary linear algebra course). It has the property that multiplying
          any vector $bf v$ by it gives the zero vector of the second vector space.



          The zero operator is a linear operator, i.e. a linear map from a vector space to a vector space (possibly the same one). It has the property that it maps any member of the first vector space to the zero vector in the second vector space.



          The zero functional is a linear functional, i.e. a linear map from a vector space to the scalars. It has the property that it maps any member of the vector space to the zero scalar.






          share|cite|improve this answer











          $endgroup$





















            2












            $begingroup$

            In an algebraic context where there is a notion of addition, $0$ is the element such that
            $$
            x + 0 = x
            $$
            for every $x$.



            If the context is the real numbers, then $0$ is just a number. If the context is the Euclidean coordinate plane, $0$ is the vector $(0,0)$. If the context is the set of real valued functions on the unit interval then $0$ is the function whose value at every point is $0$. If the context is the set of linear operators from one vector space to another then $0$ is the operator whose value at every point of the domain is the $0$ vector in the codomain.



            So the meaning of the symbol "$0$" changes depending on the context. That's potentially confusing (which is why you are asking the question.) The advantage in using the same symbol in these different contexts is that it's easy to associate that symbol with its behavior: it's the additive identity.






            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
              });


              }
              });






              Arlene is a new contributor. Be nice, and check out our Code of Conduct.










              draft saved

              draft discarded


















              StackExchange.ready(
              function () {
              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3117052%2fwhat-is-the-difference-between-a-zero-operator-zero-function-zero-scalar-and%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









              4












              $begingroup$

              The zero vector is a vector, i.e. a member of whatever vector space is under consideration. It has the property that adding it to any vector $bf v$ in the vector space leaves $bf v$ unchanged.



              The zero scalar is a scalar, i.e. a member of the field that is part of the definition of the vector space (usually the real or complex numbers in an elementary linear algebra course). It has the property that multiplying
              any vector $bf v$ by it gives the zero vector of the second vector space.



              The zero operator is a linear operator, i.e. a linear map from a vector space to a vector space (possibly the same one). It has the property that it maps any member of the first vector space to the zero vector in the second vector space.



              The zero functional is a linear functional, i.e. a linear map from a vector space to the scalars. It has the property that it maps any member of the vector space to the zero scalar.






              share|cite|improve this answer











              $endgroup$


















                4












                $begingroup$

                The zero vector is a vector, i.e. a member of whatever vector space is under consideration. It has the property that adding it to any vector $bf v$ in the vector space leaves $bf v$ unchanged.



                The zero scalar is a scalar, i.e. a member of the field that is part of the definition of the vector space (usually the real or complex numbers in an elementary linear algebra course). It has the property that multiplying
                any vector $bf v$ by it gives the zero vector of the second vector space.



                The zero operator is a linear operator, i.e. a linear map from a vector space to a vector space (possibly the same one). It has the property that it maps any member of the first vector space to the zero vector in the second vector space.



                The zero functional is a linear functional, i.e. a linear map from a vector space to the scalars. It has the property that it maps any member of the vector space to the zero scalar.






                share|cite|improve this answer











                $endgroup$
















                  4












                  4








                  4





                  $begingroup$

                  The zero vector is a vector, i.e. a member of whatever vector space is under consideration. It has the property that adding it to any vector $bf v$ in the vector space leaves $bf v$ unchanged.



                  The zero scalar is a scalar, i.e. a member of the field that is part of the definition of the vector space (usually the real or complex numbers in an elementary linear algebra course). It has the property that multiplying
                  any vector $bf v$ by it gives the zero vector of the second vector space.



                  The zero operator is a linear operator, i.e. a linear map from a vector space to a vector space (possibly the same one). It has the property that it maps any member of the first vector space to the zero vector in the second vector space.



                  The zero functional is a linear functional, i.e. a linear map from a vector space to the scalars. It has the property that it maps any member of the vector space to the zero scalar.






                  share|cite|improve this answer











                  $endgroup$



                  The zero vector is a vector, i.e. a member of whatever vector space is under consideration. It has the property that adding it to any vector $bf v$ in the vector space leaves $bf v$ unchanged.



                  The zero scalar is a scalar, i.e. a member of the field that is part of the definition of the vector space (usually the real or complex numbers in an elementary linear algebra course). It has the property that multiplying
                  any vector $bf v$ by it gives the zero vector of the second vector space.



                  The zero operator is a linear operator, i.e. a linear map from a vector space to a vector space (possibly the same one). It has the property that it maps any member of the first vector space to the zero vector in the second vector space.



                  The zero functional is a linear functional, i.e. a linear map from a vector space to the scalars. It has the property that it maps any member of the vector space to the zero scalar.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited 4 hours ago

























                  answered 4 hours ago









                  Robert IsraelRobert Israel

                  324k23213467




                  324k23213467























                      2












                      $begingroup$

                      In an algebraic context where there is a notion of addition, $0$ is the element such that
                      $$
                      x + 0 = x
                      $$
                      for every $x$.



                      If the context is the real numbers, then $0$ is just a number. If the context is the Euclidean coordinate plane, $0$ is the vector $(0,0)$. If the context is the set of real valued functions on the unit interval then $0$ is the function whose value at every point is $0$. If the context is the set of linear operators from one vector space to another then $0$ is the operator whose value at every point of the domain is the $0$ vector in the codomain.



                      So the meaning of the symbol "$0$" changes depending on the context. That's potentially confusing (which is why you are asking the question.) The advantage in using the same symbol in these different contexts is that it's easy to associate that symbol with its behavior: it's the additive identity.






                      share|cite|improve this answer









                      $endgroup$


















                        2












                        $begingroup$

                        In an algebraic context where there is a notion of addition, $0$ is the element such that
                        $$
                        x + 0 = x
                        $$
                        for every $x$.



                        If the context is the real numbers, then $0$ is just a number. If the context is the Euclidean coordinate plane, $0$ is the vector $(0,0)$. If the context is the set of real valued functions on the unit interval then $0$ is the function whose value at every point is $0$. If the context is the set of linear operators from one vector space to another then $0$ is the operator whose value at every point of the domain is the $0$ vector in the codomain.



                        So the meaning of the symbol "$0$" changes depending on the context. That's potentially confusing (which is why you are asking the question.) The advantage in using the same symbol in these different contexts is that it's easy to associate that symbol with its behavior: it's the additive identity.






                        share|cite|improve this answer









                        $endgroup$
















                          2












                          2








                          2





                          $begingroup$

                          In an algebraic context where there is a notion of addition, $0$ is the element such that
                          $$
                          x + 0 = x
                          $$
                          for every $x$.



                          If the context is the real numbers, then $0$ is just a number. If the context is the Euclidean coordinate plane, $0$ is the vector $(0,0)$. If the context is the set of real valued functions on the unit interval then $0$ is the function whose value at every point is $0$. If the context is the set of linear operators from one vector space to another then $0$ is the operator whose value at every point of the domain is the $0$ vector in the codomain.



                          So the meaning of the symbol "$0$" changes depending on the context. That's potentially confusing (which is why you are asking the question.) The advantage in using the same symbol in these different contexts is that it's easy to associate that symbol with its behavior: it's the additive identity.






                          share|cite|improve this answer









                          $endgroup$



                          In an algebraic context where there is a notion of addition, $0$ is the element such that
                          $$
                          x + 0 = x
                          $$
                          for every $x$.



                          If the context is the real numbers, then $0$ is just a number. If the context is the Euclidean coordinate plane, $0$ is the vector $(0,0)$. If the context is the set of real valued functions on the unit interval then $0$ is the function whose value at every point is $0$. If the context is the set of linear operators from one vector space to another then $0$ is the operator whose value at every point of the domain is the $0$ vector in the codomain.



                          So the meaning of the symbol "$0$" changes depending on the context. That's potentially confusing (which is why you are asking the question.) The advantage in using the same symbol in these different contexts is that it's easy to associate that symbol with its behavior: it's the additive identity.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 3 hours ago









                          Ethan BolkerEthan Bolker

                          43.4k551116




                          43.4k551116






















                              Arlene is a new contributor. Be nice, and check out our Code of Conduct.










                              draft saved

                              draft discarded


















                              Arlene is a new contributor. Be nice, and check out our Code of Conduct.













                              Arlene is a new contributor. Be nice, and check out our Code of Conduct.












                              Arlene is a new contributor. Be nice, and check out our Code of Conduct.
















                              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%2f3117052%2fwhat-is-the-difference-between-a-zero-operator-zero-function-zero-scalar-and%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

                              flock() on closed filehandle LOCK_FILE at /usr/bin/apt-mirror

                              Image
                              0 I recently upgraded an Ubuntu 16.04 virtual machine to 18.04. I regularly use this VM to run apt-mirror at school on their high-speed internet, then rsync these files to my rack-mount server at home for updating multiple computers at home without wasting the tiny amount of bandwidth I have there. Since updating I can't get apt-mirror to run on my VM. It keeps giving the error: flock() on closed filehandle LOCK_FILE at /usr/bin/apt-mirror line 214 immediately followed by: apt-mirror is already running, exiting at /usr/bin/apt-mirror line 217 apt-mirror is NOT running, and does not have a cron job. I've tried rebooting, and even looked for the old-style lockfile to delete, but I guess that method is no longer used. Is there some way to force unlock this so I can run apt-mirror? UPDATE: So... I f...
                              Read more

                              Mangá

                              Image
                                Nota:  Não confundir com Manga. Mangás e Animes Mangá Mangaká • Tankōbon Film comic La nouvelle manga Mangá original em inglês Anime História Animação influenciada por animes • Filler • OVA • Seiyū Públicos Kodomo • Shōjo • Shōnen Josei • Seinen Gêneros Mahō shōjo • Mecha • Harém Ecchi • Hentai ( Yaoi • Yuri Futanari • Shotacon • Lolicon Toddlercon ) Termos Bishōjo • Chibi • Fan service Gaiden • Kawaii • Kaitō • Light novel Mahō shōnen • Moe (antropomorfismo • Kemonomimi Nekomimi ) • Super deformed Omake • Tsundere • Yonkoma Listas Mangás • Animes • Ecchi • Mangaka • Estúdios • Revistas Fãs ( otaku ) Anime Music Video Convenções • Cosplay Dōjinshi • Fansub • Fandub • Fujoshi OS-tan • Scanlation Portal de Anime e mangá v d e O mangá (português brasileiro) ou manga (português europeu) [ 1 ] (em japonês: 漫画 , manga ? , lit. “história em quadrinhos”) , é a ...
                              Read more

                              Eduardo VII do Reino Unido

                              Image
                              Eduardo VII Rei do Reino Unido da Grã-Bretanha e Irlanda e dos Domínios Britânicos de Além-mar Imperador da Índia Rei do Reino Unido e Imperador da Índia Reinado 22 de janeiro de 1901 a 6 de maio de 1910 Coroação 9 de agosto de 1902 Delhi Durbar 1 de janeiro de 1903 Predecessora Vitória Sucessor Jorge V   Esposa Alexandra da Dinamarca Descendência Alberto Vitor, Duque de Clarence e Avondale Jorge V do Reino Unido Luísa, Princesa Real Vitória de Gales Maud de Gales Alexandre João de Gales Casa Saxe-Coburgo-Gota Nome completo Alberto Eduardo Nascimento 9 de novembro de 1841   Palácio de Buckingham, Londres, Reino Unido Morte 6 de maio de 1910 (68 anos)   Palácio de Buckingham, Londres, Reino Unido Enterro Capela de São Jorge, Windsor, Berkshire, Reino Unido 20 de maio de 1910 Pai Alberto de Saxe-Coburgo-Gota Mãe Vitória do Reino Unido Assinatura ...
                              Read more