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











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      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


















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