When does $AB$ have linearly independent columns, if $A$ and $B$ are non-square matrices?












1












$begingroup$


If




  • $A$ is $m times n$ ($m<n$), and its rows are independent


  • $B$ is $n times p$ ($p<n$), and its columns are independent


  • We also know $mge n$.



does $AB$ have linearly independent columns?



Or what additional requirements are needed for $AB$ to have linearly independent columns?






linear-algebra determinant matrix-rank







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    If




    • $A$ is $m times n$ ($m<n$), and its rows are independent


    • $B$ is $n times p$ ($p<n$), and its columns are independent


    • We also know $mge n$.



    does $AB$ have linearly independent columns?



    Or what additional requirements are needed for $AB$ to have linearly independent columns?






    linear-algebra determinant matrix-rank







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      If




      • $A$ is $m times n$ ($m<n$), and its rows are independent


      • $B$ is $n times p$ ($p<n$), and its columns are independent


      • We also know $mge n$.



      does $AB$ have linearly independent columns?



      Or what additional requirements are needed for $AB$ to have linearly independent columns?






      linear-algebra determinant matrix-rank







      share|cite|improve this question











      $endgroup$




      If




      • $A$ is $m times n$ ($m<n$), and its rows are independent


      • $B$ is $n times p$ ($p<n$), and its columns are independent


      • We also know $mge n$.



      does $AB$ have linearly independent columns?



      Or what additional requirements are needed for $AB$ to have linearly independent columns?






      linear-algebra determinant matrix-rank




      linear-algebra determinant matrix-rank






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 7 '18 at 4:26







      Oliver Chang

















      asked Dec 7 '18 at 4:06









      Oliver ChangOliver Chang

      83




      83






















          2 Answers
          2






          active

          oldest

          votes


















          1












          $begingroup$

          Since $A$ has rank $m$ and $B$ has rank $p$, $AB$ has rank at most $min(m,p)$.
          $AB$ is $mtimes p$, so it could have linearly independent columns if $m ge p$,
          but not if $m < p$.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            So it is possible that AB has linearly independent columns if $mge p$, but not always?
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:22












          • $begingroup$
            For an example with $m=p=2$ and $n=3$, try $A = pmatrix{1 & 0 & 0cr 0 & 1 & 0cr}$, $B = pmatrix{0 & 0cr 1 & 0cr 0 & 1cr}$.
            $endgroup$
            – Robert Israel
            Dec 7 '18 at 13:46



















          0












          $begingroup$

          For this we need $AB $ to have rank $p $ (since $p$ columns). All we are garanteed however is that the rank of $AB$ is less orequal to the minimum of ${m,n,p}$. So this will fail for example when $m <p$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            what if we also know $m ge p$
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:26










          • $begingroup$
            @Oliver Chang The columns should not necessarily be independent. I'd try to find an example.
            $endgroup$
            – AnyAD
            Dec 7 '18 at 4:32






          • 1




            $begingroup$
            I guess it's actually quite easy to make the rows of A orthogonal to columns of B, if n is large.
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:57











          Your Answer





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

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

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

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3029446%2fwhen-does-ab-have-linearly-independent-columns-if-a-and-b-are-non-square%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          2 Answers
          2






          active

          oldest

          votes








          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1












          $begingroup$

          Since $A$ has rank $m$ and $B$ has rank $p$, $AB$ has rank at most $min(m,p)$.
          $AB$ is $mtimes p$, so it could have linearly independent columns if $m ge p$,
          but not if $m < p$.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            So it is possible that AB has linearly independent columns if $mge p$, but not always?
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:22












          • $begingroup$
            For an example with $m=p=2$ and $n=3$, try $A = pmatrix{1 & 0 & 0cr 0 & 1 & 0cr}$, $B = pmatrix{0 & 0cr 1 & 0cr 0 & 1cr}$.
            $endgroup$
            – Robert Israel
            Dec 7 '18 at 13:46
















          1












          $begingroup$

          Since $A$ has rank $m$ and $B$ has rank $p$, $AB$ has rank at most $min(m,p)$.
          $AB$ is $mtimes p$, so it could have linearly independent columns if $m ge p$,
          but not if $m < p$.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            So it is possible that AB has linearly independent columns if $mge p$, but not always?
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:22












          • $begingroup$
            For an example with $m=p=2$ and $n=3$, try $A = pmatrix{1 & 0 & 0cr 0 & 1 & 0cr}$, $B = pmatrix{0 & 0cr 1 & 0cr 0 & 1cr}$.
            $endgroup$
            – Robert Israel
            Dec 7 '18 at 13:46














          1












          1








          1





          $begingroup$

          Since $A$ has rank $m$ and $B$ has rank $p$, $AB$ has rank at most $min(m,p)$.
          $AB$ is $mtimes p$, so it could have linearly independent columns if $m ge p$,
          but not if $m < p$.






          share|cite|improve this answer









          $endgroup$



          Since $A$ has rank $m$ and $B$ has rank $p$, $AB$ has rank at most $min(m,p)$.
          $AB$ is $mtimes p$, so it could have linearly independent columns if $m ge p$,
          but not if $m < p$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 7 '18 at 4:20









          Robert IsraelRobert Israel

          326k23215469




          326k23215469








          • 1




            $begingroup$
            So it is possible that AB has linearly independent columns if $mge p$, but not always?
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:22












          • $begingroup$
            For an example with $m=p=2$ and $n=3$, try $A = pmatrix{1 & 0 & 0cr 0 & 1 & 0cr}$, $B = pmatrix{0 & 0cr 1 & 0cr 0 & 1cr}$.
            $endgroup$
            – Robert Israel
            Dec 7 '18 at 13:46














          • 1




            $begingroup$
            So it is possible that AB has linearly independent columns if $mge p$, but not always?
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:22












          • $begingroup$
            For an example with $m=p=2$ and $n=3$, try $A = pmatrix{1 & 0 & 0cr 0 & 1 & 0cr}$, $B = pmatrix{0 & 0cr 1 & 0cr 0 & 1cr}$.
            $endgroup$
            – Robert Israel
            Dec 7 '18 at 13:46








          1




          1




          $begingroup$
          So it is possible that AB has linearly independent columns if $mge p$, but not always?
          $endgroup$
          – Oliver Chang
          Dec 7 '18 at 4:22






          $begingroup$
          So it is possible that AB has linearly independent columns if $mge p$, but not always?
          $endgroup$
          – Oliver Chang
          Dec 7 '18 at 4:22














          $begingroup$
          For an example with $m=p=2$ and $n=3$, try $A = pmatrix{1 & 0 & 0cr 0 & 1 & 0cr}$, $B = pmatrix{0 & 0cr 1 & 0cr 0 & 1cr}$.
          $endgroup$
          – Robert Israel
          Dec 7 '18 at 13:46




          $begingroup$
          For an example with $m=p=2$ and $n=3$, try $A = pmatrix{1 & 0 & 0cr 0 & 1 & 0cr}$, $B = pmatrix{0 & 0cr 1 & 0cr 0 & 1cr}$.
          $endgroup$
          – Robert Israel
          Dec 7 '18 at 13:46











          0












          $begingroup$

          For this we need $AB $ to have rank $p $ (since $p$ columns). All we are garanteed however is that the rank of $AB$ is less orequal to the minimum of ${m,n,p}$. So this will fail for example when $m <p$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            what if we also know $m ge p$
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:26










          • $begingroup$
            @Oliver Chang The columns should not necessarily be independent. I'd try to find an example.
            $endgroup$
            – AnyAD
            Dec 7 '18 at 4:32






          • 1




            $begingroup$
            I guess it's actually quite easy to make the rows of A orthogonal to columns of B, if n is large.
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:57
















          0












          $begingroup$

          For this we need $AB $ to have rank $p $ (since $p$ columns). All we are garanteed however is that the rank of $AB$ is less orequal to the minimum of ${m,n,p}$. So this will fail for example when $m <p$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            what if we also know $m ge p$
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:26










          • $begingroup$
            @Oliver Chang The columns should not necessarily be independent. I'd try to find an example.
            $endgroup$
            – AnyAD
            Dec 7 '18 at 4:32






          • 1




            $begingroup$
            I guess it's actually quite easy to make the rows of A orthogonal to columns of B, if n is large.
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:57














          0












          0








          0





          $begingroup$

          For this we need $AB $ to have rank $p $ (since $p$ columns). All we are garanteed however is that the rank of $AB$ is less orequal to the minimum of ${m,n,p}$. So this will fail for example when $m <p$.






          share|cite|improve this answer











          $endgroup$



          For this we need $AB $ to have rank $p $ (since $p$ columns). All we are garanteed however is that the rank of $AB$ is less orequal to the minimum of ${m,n,p}$. So this will fail for example when $m <p$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 7 '18 at 4:26

























          answered Dec 7 '18 at 4:24









          AnyADAnyAD

          2,098812




          2,098812












          • $begingroup$
            what if we also know $m ge p$
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:26










          • $begingroup$
            @Oliver Chang The columns should not necessarily be independent. I'd try to find an example.
            $endgroup$
            – AnyAD
            Dec 7 '18 at 4:32






          • 1




            $begingroup$
            I guess it's actually quite easy to make the rows of A orthogonal to columns of B, if n is large.
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:57


















          • $begingroup$
            what if we also know $m ge p$
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:26










          • $begingroup$
            @Oliver Chang The columns should not necessarily be independent. I'd try to find an example.
            $endgroup$
            – AnyAD
            Dec 7 '18 at 4:32






          • 1




            $begingroup$
            I guess it's actually quite easy to make the rows of A orthogonal to columns of B, if n is large.
            $endgroup$
            – Oliver Chang
            Dec 7 '18 at 4:57
















          $begingroup$
          what if we also know $m ge p$
          $endgroup$
          – Oliver Chang
          Dec 7 '18 at 4:26




          $begingroup$
          what if we also know $m ge p$
          $endgroup$
          – Oliver Chang
          Dec 7 '18 at 4:26












          $begingroup$
          @Oliver Chang The columns should not necessarily be independent. I'd try to find an example.
          $endgroup$
          – AnyAD
          Dec 7 '18 at 4:32




          $begingroup$
          @Oliver Chang The columns should not necessarily be independent. I'd try to find an example.
          $endgroup$
          – AnyAD
          Dec 7 '18 at 4:32




          1




          1




          $begingroup$
          I guess it's actually quite easy to make the rows of A orthogonal to columns of B, if n is large.
          $endgroup$
          – Oliver Chang
          Dec 7 '18 at 4:57




          $begingroup$
          I guess it's actually quite easy to make the rows of A orthogonal to columns of B, if n is large.
          $endgroup$
          – Oliver Chang
          Dec 7 '18 at 4:57


















          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%2f3029446%2fwhen-does-ab-have-linearly-independent-columns-if-a-and-b-are-non-square%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

          How to change which sound is reproduced for terminal bell?

          Image
          3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question
          Read more

          Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

          Image
          up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
          Read more

          Can I use Tabulator js library in my java Spring + Thymeleaf project?

          Image
          0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"
          Read more