Word for the dimension of the vector space in which a vector lives?












1












$begingroup$


The following issue comes up whenever I teach linear algebra: I want to have a quick way to say that a vector $(x,y,z)$ is in $mathbb{R}^3$. I am tempted to say that it has "length $3$". But then some student interprets this as saying that $sqrt{x^2+y^2+z^2} = 3$. Is there some word other than "length" I could introduce and use to consistently refer to the number of coordinates in a vector? (To point out that I am not nuts here, the commands to get this number in Mathematica and MATLAB are Length and length().)






linear-algebra vocabulary







share|improve this question











$endgroup$








  • 3




    $begingroup$
    Would "dimension" do?
    $endgroup$
    – Jasper
    Dec 9 '18 at 16:22






  • 2




    $begingroup$
    This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $mathbb{R}^{2n} simeq mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do.
    $endgroup$
    – Vandermonde
    Dec 9 '18 at 16:29










  • $begingroup$
    I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has.
    $endgroup$
    – Adam
    Dec 9 '18 at 16:44






  • 1




    $begingroup$
    @user683: $n$-vector has the unfortunate collision with certain physics terminology.
    $endgroup$
    – Willie Wong
    Dec 10 '18 at 15:02






  • 1




    $begingroup$
    More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $mathbb{F}$ with the space of $n$-tuples $mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence the length().
    $endgroup$
    – Willie Wong
    Dec 10 '18 at 15:05
















1












$begingroup$


The following issue comes up whenever I teach linear algebra: I want to have a quick way to say that a vector $(x,y,z)$ is in $mathbb{R}^3$. I am tempted to say that it has "length $3$". But then some student interprets this as saying that $sqrt{x^2+y^2+z^2} = 3$. Is there some word other than "length" I could introduce and use to consistently refer to the number of coordinates in a vector? (To point out that I am not nuts here, the commands to get this number in Mathematica and MATLAB are Length and length().)






linear-algebra vocabulary







share|improve this question











$endgroup$








  • 3




    $begingroup$
    Would "dimension" do?
    $endgroup$
    – Jasper
    Dec 9 '18 at 16:22






  • 2




    $begingroup$
    This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $mathbb{R}^{2n} simeq mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do.
    $endgroup$
    – Vandermonde
    Dec 9 '18 at 16:29










  • $begingroup$
    I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has.
    $endgroup$
    – Adam
    Dec 9 '18 at 16:44






  • 1




    $begingroup$
    @user683: $n$-vector has the unfortunate collision with certain physics terminology.
    $endgroup$
    – Willie Wong
    Dec 10 '18 at 15:02






  • 1




    $begingroup$
    More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $mathbb{F}$ with the space of $n$-tuples $mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence the length().
    $endgroup$
    – Willie Wong
    Dec 10 '18 at 15:05














1












1








1


1



$begingroup$


The following issue comes up whenever I teach linear algebra: I want to have a quick way to say that a vector $(x,y,z)$ is in $mathbb{R}^3$. I am tempted to say that it has "length $3$". But then some student interprets this as saying that $sqrt{x^2+y^2+z^2} = 3$. Is there some word other than "length" I could introduce and use to consistently refer to the number of coordinates in a vector? (To point out that I am not nuts here, the commands to get this number in Mathematica and MATLAB are Length and length().)






linear-algebra vocabulary







share|improve this question











$endgroup$




The following issue comes up whenever I teach linear algebra: I want to have a quick way to say that a vector $(x,y,z)$ is in $mathbb{R}^3$. I am tempted to say that it has "length $3$". But then some student interprets this as saying that $sqrt{x^2+y^2+z^2} = 3$. Is there some word other than "length" I could introduce and use to consistently refer to the number of coordinates in a vector? (To point out that I am not nuts here, the commands to get this number in Mathematica and MATLAB are Length and length().)






linear-algebra vocabulary




linear-algebra vocabulary






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited Dec 9 '18 at 16:38







David E Speyer

















asked Dec 9 '18 at 15:42









David E SpeyerDavid E Speyer

2,305920




2,305920








  • 3




    $begingroup$
    Would "dimension" do?
    $endgroup$
    – Jasper
    Dec 9 '18 at 16:22






  • 2




    $begingroup$
    This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $mathbb{R}^{2n} simeq mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do.
    $endgroup$
    – Vandermonde
    Dec 9 '18 at 16:29










  • $begingroup$
    I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has.
    $endgroup$
    – Adam
    Dec 9 '18 at 16:44






  • 1




    $begingroup$
    @user683: $n$-vector has the unfortunate collision with certain physics terminology.
    $endgroup$
    – Willie Wong
    Dec 10 '18 at 15:02






  • 1




    $begingroup$
    More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $mathbb{F}$ with the space of $n$-tuples $mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence the length().
    $endgroup$
    – Willie Wong
    Dec 10 '18 at 15:05














  • 3




    $begingroup$
    Would "dimension" do?
    $endgroup$
    – Jasper
    Dec 9 '18 at 16:22






  • 2




    $begingroup$
    This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $mathbb{R}^{2n} simeq mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do.
    $endgroup$
    – Vandermonde
    Dec 9 '18 at 16:29










  • $begingroup$
    I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has.
    $endgroup$
    – Adam
    Dec 9 '18 at 16:44






  • 1




    $begingroup$
    @user683: $n$-vector has the unfortunate collision with certain physics terminology.
    $endgroup$
    – Willie Wong
    Dec 10 '18 at 15:02






  • 1




    $begingroup$
    More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $mathbb{F}$ with the space of $n$-tuples $mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence the length().
    $endgroup$
    – Willie Wong
    Dec 10 '18 at 15:05








3




3




$begingroup$
Would "dimension" do?
$endgroup$
– Jasper
Dec 9 '18 at 16:22




$begingroup$
Would "dimension" do?
$endgroup$
– Jasper
Dec 9 '18 at 16:22




2




2




$begingroup$
This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $mathbb{R}^{2n} simeq mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do.
$endgroup$
– Vandermonde
Dec 9 '18 at 16:29




$begingroup$
This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $mathbb{R}^{2n} simeq mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do.
$endgroup$
– Vandermonde
Dec 9 '18 at 16:29












$begingroup$
I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has.
$endgroup$
– Adam
Dec 9 '18 at 16:44




$begingroup$
I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has.
$endgroup$
– Adam
Dec 9 '18 at 16:44




1




1




$begingroup$
@user683: $n$-vector has the unfortunate collision with certain physics terminology.
$endgroup$
– Willie Wong
Dec 10 '18 at 15:02




$begingroup$
@user683: $n$-vector has the unfortunate collision with certain physics terminology.
$endgroup$
– Willie Wong
Dec 10 '18 at 15:02




1




1




$begingroup$
More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $mathbb{F}$ with the space of $n$-tuples $mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence the length().
$endgroup$
– Willie Wong
Dec 10 '18 at 15:05




$begingroup$
More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $mathbb{F}$ with the space of $n$-tuples $mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence the length().
$endgroup$
– Willie Wong
Dec 10 '18 at 15:05










3 Answers
3






active

oldest

votes


















5












$begingroup$

If all your spaces are $mathbb{R}^n$ then you can say n-dimensional (three-dimensional for $mathbb{R}^3$). However the space of matrices $displaystyle left(begin{array}{cc}a&b\c&dend{array}right)$ ($a,b,c,dinmathbb{C}$) is four-dimensional over $mathbb{C}$, eight-dimensional over $mathbb{R}$ and infinite-dimensional over $mathbb{Q}$. So be clear what the base field is. Alternatively you could just write $(a,b,c)inleft(mathbb{R}^3,+,cdotright)$ or more briefly $(a,b,c)inmathbb{R}^3$.






share|improve this answer









$endgroup$





















    5












    $begingroup$

    Along with "dimension", you could also use "component".
    A vector in three dimensions has three components.






    share|improve this answer









    $endgroup$





















      4












      $begingroup$

      I say "This is a 3D vector" or "This is a 7D vector".






      share|improve this answer









      $endgroup$













      • $begingroup$
        I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
        $endgroup$
        – Steven Gubkin
        Dec 11 '18 at 1:41











      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: "548"
      };
      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: false,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: null,
      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%2fmatheducators.stackexchange.com%2fquestions%2f14882%2fword-for-the-dimension-of-the-vector-space-in-which-a-vector-lives%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      5












      $begingroup$

      If all your spaces are $mathbb{R}^n$ then you can say n-dimensional (three-dimensional for $mathbb{R}^3$). However the space of matrices $displaystyle left(begin{array}{cc}a&b\c&dend{array}right)$ ($a,b,c,dinmathbb{C}$) is four-dimensional over $mathbb{C}$, eight-dimensional over $mathbb{R}$ and infinite-dimensional over $mathbb{Q}$. So be clear what the base field is. Alternatively you could just write $(a,b,c)inleft(mathbb{R}^3,+,cdotright)$ or more briefly $(a,b,c)inmathbb{R}^3$.






      share|improve this answer









      $endgroup$


















        5












        $begingroup$

        If all your spaces are $mathbb{R}^n$ then you can say n-dimensional (three-dimensional for $mathbb{R}^3$). However the space of matrices $displaystyle left(begin{array}{cc}a&b\c&dend{array}right)$ ($a,b,c,dinmathbb{C}$) is four-dimensional over $mathbb{C}$, eight-dimensional over $mathbb{R}$ and infinite-dimensional over $mathbb{Q}$. So be clear what the base field is. Alternatively you could just write $(a,b,c)inleft(mathbb{R}^3,+,cdotright)$ or more briefly $(a,b,c)inmathbb{R}^3$.






        share|improve this answer









        $endgroup$
















          5












          5








          5





          $begingroup$

          If all your spaces are $mathbb{R}^n$ then you can say n-dimensional (three-dimensional for $mathbb{R}^3$). However the space of matrices $displaystyle left(begin{array}{cc}a&b\c&dend{array}right)$ ($a,b,c,dinmathbb{C}$) is four-dimensional over $mathbb{C}$, eight-dimensional over $mathbb{R}$ and infinite-dimensional over $mathbb{Q}$. So be clear what the base field is. Alternatively you could just write $(a,b,c)inleft(mathbb{R}^3,+,cdotright)$ or more briefly $(a,b,c)inmathbb{R}^3$.






          share|improve this answer









          $endgroup$



          If all your spaces are $mathbb{R}^n$ then you can say n-dimensional (three-dimensional for $mathbb{R}^3$). However the space of matrices $displaystyle left(begin{array}{cc}a&b\c&dend{array}right)$ ($a,b,c,dinmathbb{C}$) is four-dimensional over $mathbb{C}$, eight-dimensional over $mathbb{R}$ and infinite-dimensional over $mathbb{Q}$. So be clear what the base field is. Alternatively you could just write $(a,b,c)inleft(mathbb{R}^3,+,cdotright)$ or more briefly $(a,b,c)inmathbb{R}^3$.







          share|improve this answer












          share|improve this answer



          share|improve this answer










          answered Dec 9 '18 at 17:45









          BPPBPP

          576316




          576316























              5












              $begingroup$

              Along with "dimension", you could also use "component".
              A vector in three dimensions has three components.






              share|improve this answer









              $endgroup$


















                5












                $begingroup$

                Along with "dimension", you could also use "component".
                A vector in three dimensions has three components.






                share|improve this answer









                $endgroup$
















                  5












                  5








                  5





                  $begingroup$

                  Along with "dimension", you could also use "component".
                  A vector in three dimensions has three components.






                  share|improve this answer









                  $endgroup$



                  Along with "dimension", you could also use "component".
                  A vector in three dimensions has three components.







                  share|improve this answer












                  share|improve this answer



                  share|improve this answer










                  answered Dec 9 '18 at 21:59









                  robphyrobphy

                  3713




                  3713























                      4












                      $begingroup$

                      I say "This is a 3D vector" or "This is a 7D vector".






                      share|improve this answer









                      $endgroup$













                      • $begingroup$
                        I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
                        $endgroup$
                        – Steven Gubkin
                        Dec 11 '18 at 1:41
















                      4












                      $begingroup$

                      I say "This is a 3D vector" or "This is a 7D vector".






                      share|improve this answer









                      $endgroup$













                      • $begingroup$
                        I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
                        $endgroup$
                        – Steven Gubkin
                        Dec 11 '18 at 1:41














                      4












                      4








                      4





                      $begingroup$

                      I say "This is a 3D vector" or "This is a 7D vector".






                      share|improve this answer









                      $endgroup$



                      I say "This is a 3D vector" or "This is a 7D vector".







                      share|improve this answer












                      share|improve this answer



                      share|improve this answer










                      answered Dec 9 '18 at 16:34









                      Steven GubkinSteven Gubkin

                      8,42812348




                      8,42812348












                      • $begingroup$
                        I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
                        $endgroup$
                        – Steven Gubkin
                        Dec 11 '18 at 1:41


















                      • $begingroup$
                        I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
                        $endgroup$
                        – Steven Gubkin
                        Dec 11 '18 at 1:41
















                      $begingroup$
                      I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
                      $endgroup$
                      – Steven Gubkin
                      Dec 11 '18 at 1:41




                      $begingroup$
                      I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space.
                      $endgroup$
                      – Steven Gubkin
                      Dec 11 '18 at 1:41


















                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Mathematics Educators 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%2fmatheducators.stackexchange.com%2fquestions%2f14882%2fword-for-the-dimension-of-the-vector-space-in-which-a-vector-lives%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

                      Different font size/position of beamer's navigation symbols template's content depending on regular/plain...

                      Image
                      3 Let's suppose I'd like to divert the navigation symbols template (e.g. in order replace them by the frame numbers, see this bug report ;) I would be facing troubles regarding the font size/position, depending on regular/plain frames, as shown by the following MCE. How could I get rid of these troubles? documentclass{beamer} usetheme{PaloAlto} setbeamertemplate{navigation symbols}{insertframenumber/inserttotalframenumber} begin{document} begin{frame} frametitle{A regular frame} end{frame} begin{frame}[plain] frametitle{A plain frame} end{frame} end{document} beamer header-footer share | improve this question asked Dec 10 '18 a
                      Read more

                      Berounka

                      Image
                      Berounka.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Caractéristiques Longueur 139  km Bassin 8 861  km 2 Bassin collecteur Elbe Débit moyen ? Cours Source Confluence de la Mže et de la Radbuza · Localisation Pilsen · Altitude 305  m · Coordonnées 49° 45′ 13″ N, 13° 23′ 24″ E Confluence Vltava · Localisation Prague · Altitude 187  m · Coordonnées 49° 59′ 43″ N, 14° 24′ 04″ E Géographie Pays traversés   République tchèque modifier   La Berounka est une rivière de la République tchèque. Elle porte le nom de Mže depuis sa source en Allemagne près de la frontière jusqu'à sa confluence avec la Radbuza à Plzeň. Elle continue ensuite sous le nom de Berounka jusqu'à sa confluence avec la Vltava, près de Prague. La rivière est une destination prisée pour les amateurs de canoë, qui appréc
                      Read more

                      I want to find a topological embedding $f : X rightarrow Y$ and $g: Y rightarrow X$, yet $X$ is not...

                      Image
                      0 0 $begingroup$ This is related to a problem of showing that none of the intervals $(0,1], (0,1), [0,1]$ are homeomorphic to another, since it's in the same problem block. I've tried using $f(x) = 1 - x , f(x) = 1/(x-1)$ between varying pairs of those intervals to no avail. Do I need a function as complicated as the topologist's sine to show this? general-topology connectedness share | cite | improve this question edited Dec 10 '18 at 3:46 user1101010 775 1 7 30 asked Sep 24 '13 at 23:15
                      Read more