A set of vectors in $R^2$ for which $x + y$ stays in the set but $frac12x$ may be outside.












3












$begingroup$


These questions refer to subspace.



One requirement can be met while the other fails. Show this by finding

(a) A set of vectors in $R^2$ for which $x + y$ stays in the set but $frac12x$ may be outside.

(b) A set of vectors in $R^2$ (other than two quarter-planes) for which every $cx$ stays in the set but $x + y$ may be outside.



these are the answers my textbook gives,

(a) The vectors with integer components allow addition, but not multiplication by $frac12$

This is saying that you can't multiply a vector by a non-integer? why not?



(b) Remove the x axis from the xy plane (but leave the origin). Multiplication by any c is allowed but not all vector additions.

If you remove the x-axis then you are only left with vectors on the y-axis, so shouldn't it still be closed under addition?






linear-algebra







share|cite|improve this question









$endgroup$

















    3












    $begingroup$


    These questions refer to subspace.



    One requirement can be met while the other fails. Show this by finding

    (a) A set of vectors in $R^2$ for which $x + y$ stays in the set but $frac12x$ may be outside.

    (b) A set of vectors in $R^2$ (other than two quarter-planes) for which every $cx$ stays in the set but $x + y$ may be outside.



    these are the answers my textbook gives,

    (a) The vectors with integer components allow addition, but not multiplication by $frac12$

    This is saying that you can't multiply a vector by a non-integer? why not?



    (b) Remove the x axis from the xy plane (but leave the origin). Multiplication by any c is allowed but not all vector additions.

    If you remove the x-axis then you are only left with vectors on the y-axis, so shouldn't it still be closed under addition?






    linear-algebra







    share|cite|improve this question









    $endgroup$















      3












      3








      3





      $begingroup$


      These questions refer to subspace.



      One requirement can be met while the other fails. Show this by finding

      (a) A set of vectors in $R^2$ for which $x + y$ stays in the set but $frac12x$ may be outside.

      (b) A set of vectors in $R^2$ (other than two quarter-planes) for which every $cx$ stays in the set but $x + y$ may be outside.



      these are the answers my textbook gives,

      (a) The vectors with integer components allow addition, but not multiplication by $frac12$

      This is saying that you can't multiply a vector by a non-integer? why not?



      (b) Remove the x axis from the xy plane (but leave the origin). Multiplication by any c is allowed but not all vector additions.

      If you remove the x-axis then you are only left with vectors on the y-axis, so shouldn't it still be closed under addition?






      linear-algebra







      share|cite|improve this question









      $endgroup$




      These questions refer to subspace.



      One requirement can be met while the other fails. Show this by finding

      (a) A set of vectors in $R^2$ for which $x + y$ stays in the set but $frac12x$ may be outside.

      (b) A set of vectors in $R^2$ (other than two quarter-planes) for which every $cx$ stays in the set but $x + y$ may be outside.



      these are the answers my textbook gives,

      (a) The vectors with integer components allow addition, but not multiplication by $frac12$

      This is saying that you can't multiply a vector by a non-integer? why not?



      (b) Remove the x axis from the xy plane (but leave the origin). Multiplication by any c is allowed but not all vector additions.

      If you remove the x-axis then you are only left with vectors on the y-axis, so shouldn't it still be closed under addition?






      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 Feb 26 '16 at 22:21









      idknuttinidknuttin

      1,2071631




      1,2071631






















          4 Answers
          4






          active

          oldest

          votes


















          1












          $begingroup$


          1. $frac12(1, 1)$ is not an integer vector.

          2. The plane with the $x$-axis removed contains all points with the first coordinates non-zero.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            a set of integer vectors can be considered a subset?
            $endgroup$
            – idknuttin
            Feb 26 '16 at 22:26










          • $begingroup$
            @idknuttin ??? If it is a set, why would it not be a subset?
            $endgroup$
            – Igor Rivin
            Feb 26 '16 at 22:27










          • $begingroup$
            I meant a set of integer vectors can be considered a subspace?
            $endgroup$
            – idknuttin
            Feb 26 '16 at 22:30










          • $begingroup$
            @idknuttin No, but it does not say it should be a subspace.
            $endgroup$
            – Igor Rivin
            Feb 26 '16 at 22:30



















          2












          $begingroup$

          (a) If you multiply a nonzero vector with integer components by a noninteger, the result is not in "the set of vectors with integer components" anymore.



          (b) if you remove the $x$-axis, you still have points like $(1,1)$ and $(1,-1)$ even though you don't have their sum.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Technically scaling by 0 makes b) not work.
            $endgroup$
            – Improve
            Feb 26 '16 at 22:44












          • $begingroup$
            @improve since the book said "but leave the origin" and the question was about closure under addition, I didn't think that was worth mentioning here. But you're certainly correct.
            $endgroup$
            – Mark S.
            Feb 26 '16 at 22:46










          • $begingroup$
            I did not read the entire question.
            $endgroup$
            – Improve
            Feb 26 '16 at 22:47





















          1












          $begingroup$

          They are asking for sets that satisfy some property of vector spaces, but that are not vector spaces.



          (a) the only operation they are allowing is addition, thus clearly vectors with entire componends only add to vectors with entire components, it's not a vector space since $mathbb{Z}$ is not a field (you don't have $n^{-1}$.



          (b) You are assuming it's a vector space, it is not, thats why the set doesn't collapses to only the y-axis.






          share|cite|improve this answer









          $endgroup$





















            0












            $begingroup$

            b)



            imagine a line that is exactly on the y-axis.
            you have a vector v = (0,1) , so cv always on the y-axis but for example
            (0,1) + (1,1) = (1,2) wich is not on the line (y-axix)






            share|cite|improve this answer









            $endgroup$














              Your Answer








              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%2f1673698%2fa-set-of-vectors-in-r2-for-which-x-y-stays-in-the-set-but-frac12x-may%23new-answer', 'question_page');
              }
              );

              Post as a guest















              Required, but never shown

























              4 Answers
              4






              active

              oldest

              votes








              4 Answers
              4






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              1












              $begingroup$


              1. $frac12(1, 1)$ is not an integer vector.

              2. The plane with the $x$-axis removed contains all points with the first coordinates non-zero.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                a set of integer vectors can be considered a subset?
                $endgroup$
                – idknuttin
                Feb 26 '16 at 22:26










              • $begingroup$
                @idknuttin ??? If it is a set, why would it not be a subset?
                $endgroup$
                – Igor Rivin
                Feb 26 '16 at 22:27










              • $begingroup$
                I meant a set of integer vectors can be considered a subspace?
                $endgroup$
                – idknuttin
                Feb 26 '16 at 22:30










              • $begingroup$
                @idknuttin No, but it does not say it should be a subspace.
                $endgroup$
                – Igor Rivin
                Feb 26 '16 at 22:30
















              1












              $begingroup$


              1. $frac12(1, 1)$ is not an integer vector.

              2. The plane with the $x$-axis removed contains all points with the first coordinates non-zero.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                a set of integer vectors can be considered a subset?
                $endgroup$
                – idknuttin
                Feb 26 '16 at 22:26










              • $begingroup$
                @idknuttin ??? If it is a set, why would it not be a subset?
                $endgroup$
                – Igor Rivin
                Feb 26 '16 at 22:27










              • $begingroup$
                I meant a set of integer vectors can be considered a subspace?
                $endgroup$
                – idknuttin
                Feb 26 '16 at 22:30










              • $begingroup$
                @idknuttin No, but it does not say it should be a subspace.
                $endgroup$
                – Igor Rivin
                Feb 26 '16 at 22:30














              1












              1








              1





              $begingroup$


              1. $frac12(1, 1)$ is not an integer vector.

              2. The plane with the $x$-axis removed contains all points with the first coordinates non-zero.






              share|cite|improve this answer









              $endgroup$




              1. $frac12(1, 1)$ is not an integer vector.

              2. The plane with the $x$-axis removed contains all points with the first coordinates non-zero.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Feb 26 '16 at 22:24









              Igor RivinIgor Rivin

              16.1k11234




              16.1k11234












              • $begingroup$
                a set of integer vectors can be considered a subset?
                $endgroup$
                – idknuttin
                Feb 26 '16 at 22:26










              • $begingroup$
                @idknuttin ??? If it is a set, why would it not be a subset?
                $endgroup$
                – Igor Rivin
                Feb 26 '16 at 22:27










              • $begingroup$
                I meant a set of integer vectors can be considered a subspace?
                $endgroup$
                – idknuttin
                Feb 26 '16 at 22:30










              • $begingroup$
                @idknuttin No, but it does not say it should be a subspace.
                $endgroup$
                – Igor Rivin
                Feb 26 '16 at 22:30


















              • $begingroup$
                a set of integer vectors can be considered a subset?
                $endgroup$
                – idknuttin
                Feb 26 '16 at 22:26










              • $begingroup$
                @idknuttin ??? If it is a set, why would it not be a subset?
                $endgroup$
                – Igor Rivin
                Feb 26 '16 at 22:27










              • $begingroup$
                I meant a set of integer vectors can be considered a subspace?
                $endgroup$
                – idknuttin
                Feb 26 '16 at 22:30










              • $begingroup$
                @idknuttin No, but it does not say it should be a subspace.
                $endgroup$
                – Igor Rivin
                Feb 26 '16 at 22:30
















              $begingroup$
              a set of integer vectors can be considered a subset?
              $endgroup$
              – idknuttin
              Feb 26 '16 at 22:26




              $begingroup$
              a set of integer vectors can be considered a subset?
              $endgroup$
              – idknuttin
              Feb 26 '16 at 22:26












              $begingroup$
              @idknuttin ??? If it is a set, why would it not be a subset?
              $endgroup$
              – Igor Rivin
              Feb 26 '16 at 22:27




              $begingroup$
              @idknuttin ??? If it is a set, why would it not be a subset?
              $endgroup$
              – Igor Rivin
              Feb 26 '16 at 22:27












              $begingroup$
              I meant a set of integer vectors can be considered a subspace?
              $endgroup$
              – idknuttin
              Feb 26 '16 at 22:30




              $begingroup$
              I meant a set of integer vectors can be considered a subspace?
              $endgroup$
              – idknuttin
              Feb 26 '16 at 22:30












              $begingroup$
              @idknuttin No, but it does not say it should be a subspace.
              $endgroup$
              – Igor Rivin
              Feb 26 '16 at 22:30




              $begingroup$
              @idknuttin No, but it does not say it should be a subspace.
              $endgroup$
              – Igor Rivin
              Feb 26 '16 at 22:30











              2












              $begingroup$

              (a) If you multiply a nonzero vector with integer components by a noninteger, the result is not in "the set of vectors with integer components" anymore.



              (b) if you remove the $x$-axis, you still have points like $(1,1)$ and $(1,-1)$ even though you don't have their sum.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                Technically scaling by 0 makes b) not work.
                $endgroup$
                – Improve
                Feb 26 '16 at 22:44












              • $begingroup$
                @improve since the book said "but leave the origin" and the question was about closure under addition, I didn't think that was worth mentioning here. But you're certainly correct.
                $endgroup$
                – Mark S.
                Feb 26 '16 at 22:46










              • $begingroup$
                I did not read the entire question.
                $endgroup$
                – Improve
                Feb 26 '16 at 22:47


















              2












              $begingroup$

              (a) If you multiply a nonzero vector with integer components by a noninteger, the result is not in "the set of vectors with integer components" anymore.



              (b) if you remove the $x$-axis, you still have points like $(1,1)$ and $(1,-1)$ even though you don't have their sum.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                Technically scaling by 0 makes b) not work.
                $endgroup$
                – Improve
                Feb 26 '16 at 22:44












              • $begingroup$
                @improve since the book said "but leave the origin" and the question was about closure under addition, I didn't think that was worth mentioning here. But you're certainly correct.
                $endgroup$
                – Mark S.
                Feb 26 '16 at 22:46










              • $begingroup$
                I did not read the entire question.
                $endgroup$
                – Improve
                Feb 26 '16 at 22:47
















              2












              2








              2





              $begingroup$

              (a) If you multiply a nonzero vector with integer components by a noninteger, the result is not in "the set of vectors with integer components" anymore.



              (b) if you remove the $x$-axis, you still have points like $(1,1)$ and $(1,-1)$ even though you don't have their sum.






              share|cite|improve this answer









              $endgroup$



              (a) If you multiply a nonzero vector with integer components by a noninteger, the result is not in "the set of vectors with integer components" anymore.



              (b) if you remove the $x$-axis, you still have points like $(1,1)$ and $(1,-1)$ even though you don't have their sum.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Feb 26 '16 at 22:26









              Mark S.Mark S.

              12.5k22772




              12.5k22772












              • $begingroup$
                Technically scaling by 0 makes b) not work.
                $endgroup$
                – Improve
                Feb 26 '16 at 22:44












              • $begingroup$
                @improve since the book said "but leave the origin" and the question was about closure under addition, I didn't think that was worth mentioning here. But you're certainly correct.
                $endgroup$
                – Mark S.
                Feb 26 '16 at 22:46










              • $begingroup$
                I did not read the entire question.
                $endgroup$
                – Improve
                Feb 26 '16 at 22:47




















              • $begingroup$
                Technically scaling by 0 makes b) not work.
                $endgroup$
                – Improve
                Feb 26 '16 at 22:44












              • $begingroup$
                @improve since the book said "but leave the origin" and the question was about closure under addition, I didn't think that was worth mentioning here. But you're certainly correct.
                $endgroup$
                – Mark S.
                Feb 26 '16 at 22:46










              • $begingroup$
                I did not read the entire question.
                $endgroup$
                – Improve
                Feb 26 '16 at 22:47


















              $begingroup$
              Technically scaling by 0 makes b) not work.
              $endgroup$
              – Improve
              Feb 26 '16 at 22:44






              $begingroup$
              Technically scaling by 0 makes b) not work.
              $endgroup$
              – Improve
              Feb 26 '16 at 22:44














              $begingroup$
              @improve since the book said "but leave the origin" and the question was about closure under addition, I didn't think that was worth mentioning here. But you're certainly correct.
              $endgroup$
              – Mark S.
              Feb 26 '16 at 22:46




              $begingroup$
              @improve since the book said "but leave the origin" and the question was about closure under addition, I didn't think that was worth mentioning here. But you're certainly correct.
              $endgroup$
              – Mark S.
              Feb 26 '16 at 22:46












              $begingroup$
              I did not read the entire question.
              $endgroup$
              – Improve
              Feb 26 '16 at 22:47






              $begingroup$
              I did not read the entire question.
              $endgroup$
              – Improve
              Feb 26 '16 at 22:47













              1












              $begingroup$

              They are asking for sets that satisfy some property of vector spaces, but that are not vector spaces.



              (a) the only operation they are allowing is addition, thus clearly vectors with entire componends only add to vectors with entire components, it's not a vector space since $mathbb{Z}$ is not a field (you don't have $n^{-1}$.



              (b) You are assuming it's a vector space, it is not, thats why the set doesn't collapses to only the y-axis.






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                They are asking for sets that satisfy some property of vector spaces, but that are not vector spaces.



                (a) the only operation they are allowing is addition, thus clearly vectors with entire componends only add to vectors with entire components, it's not a vector space since $mathbb{Z}$ is not a field (you don't have $n^{-1}$.



                (b) You are assuming it's a vector space, it is not, thats why the set doesn't collapses to only the y-axis.






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  They are asking for sets that satisfy some property of vector spaces, but that are not vector spaces.



                  (a) the only operation they are allowing is addition, thus clearly vectors with entire componends only add to vectors with entire components, it's not a vector space since $mathbb{Z}$ is not a field (you don't have $n^{-1}$.



                  (b) You are assuming it's a vector space, it is not, thats why the set doesn't collapses to only the y-axis.






                  share|cite|improve this answer









                  $endgroup$



                  They are asking for sets that satisfy some property of vector spaces, but that are not vector spaces.



                  (a) the only operation they are allowing is addition, thus clearly vectors with entire componends only add to vectors with entire components, it's not a vector space since $mathbb{Z}$ is not a field (you don't have $n^{-1}$.



                  (b) You are assuming it's a vector space, it is not, thats why the set doesn't collapses to only the y-axis.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Feb 26 '16 at 22:29









                  panchopancho

                  573312




                  573312























                      0












                      $begingroup$

                      b)



                      imagine a line that is exactly on the y-axis.
                      you have a vector v = (0,1) , so cv always on the y-axis but for example
                      (0,1) + (1,1) = (1,2) wich is not on the line (y-axix)






                      share|cite|improve this answer









                      $endgroup$


















                        0












                        $begingroup$

                        b)



                        imagine a line that is exactly on the y-axis.
                        you have a vector v = (0,1) , so cv always on the y-axis but for example
                        (0,1) + (1,1) = (1,2) wich is not on the line (y-axix)






                        share|cite|improve this answer









                        $endgroup$
















                          0












                          0








                          0





                          $begingroup$

                          b)



                          imagine a line that is exactly on the y-axis.
                          you have a vector v = (0,1) , so cv always on the y-axis but for example
                          (0,1) + (1,1) = (1,2) wich is not on the line (y-axix)






                          share|cite|improve this answer









                          $endgroup$



                          b)



                          imagine a line that is exactly on the y-axis.
                          you have a vector v = (0,1) , so cv always on the y-axis but for example
                          (0,1) + (1,1) = (1,2) wich is not on the line (y-axix)







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Dec 24 '18 at 18:59









                          Bishoy AbdBishoy Abd

                          1012




                          1012






























                              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%2f1673698%2fa-set-of-vectors-in-r2-for-which-x-y-stays-in-the-set-but-frac12x-may%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

                              Plaza Victoria

                              Image
                              Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l
                              Read more

                              Puebla de Zaragoza

                              Image
                              Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc
                              Read more

                              Musa

                              Image
                              Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo
                              Read more