question about determinant variety in Karen Smith's “An Invitation to Algebraic Geometry”

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
2
down vote

favorite
1












I don't understand in the example below:



If $k geq n$, then the determinant variety is the whole space ? enter image description here




linear-algebra algebraic-geometry




share|cite|improve this question





















  • She is talking about $n times n$-matrices. If $k geq n$, then each $n times n$-matrix has rank at most $k$.
    – darij grinberg
    Jul 22 at 18:15














up vote
2
down vote

favorite
1












I don't understand in the example below:



If $k geq n$, then the determinant variety is the whole space ? enter image description here




linear-algebra algebraic-geometry




share|cite|improve this question





















  • She is talking about $n times n$-matrices. If $k geq n$, then each $n times n$-matrix has rank at most $k$.
    – darij grinberg
    Jul 22 at 18:15












up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





I don't understand in the example below:



If $k geq n$, then the determinant variety is the whole space ? enter image description here




linear-algebra algebraic-geometry




share|cite|improve this question













I don't understand in the example below:



If $k geq n$, then the determinant variety is the whole space ? enter image description here





linear-algebra algebraic-geometry





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 22 at 16:38
























asked Jul 22 at 16:20









Newbie

384111




384111











  • She is talking about $n times n$-matrices. If $k geq n$, then each $n times n$-matrix has rank at most $k$.
    – darij grinberg
    Jul 22 at 18:15
















  • She is talking about $n times n$-matrices. If $k geq n$, then each $n times n$-matrix has rank at most $k$.
    – darij grinberg
    Jul 22 at 18:15















She is talking about $n times n$-matrices. If $k geq n$, then each $n times n$-matrix has rank at most $k$.
– darij grinberg
Jul 22 at 18:15




She is talking about $n times n$-matrices. If $k geq n$, then each $n times n$-matrix has rank at most $k$.
– darij grinberg
Jul 22 at 18:15










2 Answers
2






active

oldest

votes

















up vote
4
down vote



accepted










An $n times n$ matrix has $n$ rows and $n$ columns. The rank of a matrix is the number of linearly independent rows or columns (row rank and column rank being the same). The greatest possible number of linearly independent vectors out of a set of $n$ vectors is obviously $n$. Thus for $k ge n$, every $n times n$ matrix $A$ satisfies



$textrank(A) le n le k. tag 1$






share|cite|improve this answer




























    up vote
    3
    down vote













    Every $n times n$ matrix has rank at most $n$.






    share|cite|improve this answer





















      Your Answer




      StackExchange.ifUsing("editor", function ()
      return StackExchange.using("mathjaxEditing", function ()
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      );
      );
      , "mathjax-editing");

      StackExchange.ready(function()
      var channelOptions =
      tags: "".split(" "),
      id: "69"
      ;
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function()
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled)
      StackExchange.using("snippets", function()
      createEditor();
      );

      else
      createEditor();

      );

      function createEditor()
      StackExchange.prepareEditor(
      heartbeatType: 'answer',
      convertImagesToLinks: true,
      noModals: false,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      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%2f2859540%2fquestion-about-determinant-variety-in-karen-smiths-an-invitation-to-algebraic%23new-answer', 'question_page');

      );

      Post as a guest

















      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      4
      down vote



      accepted










      An $n times n$ matrix has $n$ rows and $n$ columns. The rank of a matrix is the number of linearly independent rows or columns (row rank and column rank being the same). The greatest possible number of linearly independent vectors out of a set of $n$ vectors is obviously $n$. Thus for $k ge n$, every $n times n$ matrix $A$ satisfies



      $textrank(A) le n le k. tag 1$






      share|cite|improve this answer

























        up vote
        4
        down vote



        accepted










        An $n times n$ matrix has $n$ rows and $n$ columns. The rank of a matrix is the number of linearly independent rows or columns (row rank and column rank being the same). The greatest possible number of linearly independent vectors out of a set of $n$ vectors is obviously $n$. Thus for $k ge n$, every $n times n$ matrix $A$ satisfies



        $textrank(A) le n le k. tag 1$






        share|cite|improve this answer























          up vote
          4
          down vote



          accepted







          up vote
          4
          down vote



          accepted






          An $n times n$ matrix has $n$ rows and $n$ columns. The rank of a matrix is the number of linearly independent rows or columns (row rank and column rank being the same). The greatest possible number of linearly independent vectors out of a set of $n$ vectors is obviously $n$. Thus for $k ge n$, every $n times n$ matrix $A$ satisfies



          $textrank(A) le n le k. tag 1$






          share|cite|improve this answer













          An $n times n$ matrix has $n$ rows and $n$ columns. The rank of a matrix is the number of linearly independent rows or columns (row rank and column rank being the same). The greatest possible number of linearly independent vectors out of a set of $n$ vectors is obviously $n$. Thus for $k ge n$, every $n times n$ matrix $A$ satisfies



          $textrank(A) le n le k. tag 1$







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Jul 22 at 16:31









          Robert Lewis

          36.9k22255




          36.9k22255




















              up vote
              3
              down vote













              Every $n times n$ matrix has rank at most $n$.






              share|cite|improve this answer

























                up vote
                3
                down vote













                Every $n times n$ matrix has rank at most $n$.






                share|cite|improve this answer























                  up vote
                  3
                  down vote










                  up vote
                  3
                  down vote









                  Every $n times n$ matrix has rank at most $n$.






                  share|cite|improve this answer













                  Every $n times n$ matrix has rank at most $n$.







                  share|cite|improve this answer













                  share|cite|improve this answer



                  share|cite|improve this answer











                  answered Jul 22 at 16:26









                  Hurkyl

                  108k9112253




                  108k9112253






















                       

                      draft saved


                      draft discarded


























                       


                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function ()
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2859540%2fquestion-about-determinant-variety-in-karen-smiths-an-invitation-to-algebraic%23new-answer', 'question_page');

                      );

                      Post as a guest
































































                      Comments

                      Post a Comment

                      Popular posts from this blog

                      What is the equation of a 3D cone with generalised tilt?

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
                      Read more

                      Relationship between determinant of matrix and determinant of adjoint?

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
                      Read more

                      Color the edges and diagonals of a regular polygon

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
                      Read more