Null space and related question

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











up vote
-1
down vote

favorite












Suppose vectors $v_i, i=1, ..., p$ is a basis of the null space of $mtimes n$ matrix $B$. If $xin mathbbR^n, <x, v_i>=0, i=1, ..., p$, then there exists a vector $y$ such that



$x=B^intercal y$.



How to prove this assertion if valid? I haven't find out a clear path.










share|cite|improve this question

























    up vote
    -1
    down vote

    favorite












    Suppose vectors $v_i, i=1, ..., p$ is a basis of the null space of $mtimes n$ matrix $B$. If $xin mathbbR^n, <x, v_i>=0, i=1, ..., p$, then there exists a vector $y$ such that



    $x=B^intercal y$.



    How to prove this assertion if valid? I haven't find out a clear path.










    share|cite|improve this question























      up vote
      -1
      down vote

      favorite









      up vote
      -1
      down vote

      favorite











      Suppose vectors $v_i, i=1, ..., p$ is a basis of the null space of $mtimes n$ matrix $B$. If $xin mathbbR^n, <x, v_i>=0, i=1, ..., p$, then there exists a vector $y$ such that



      $x=B^intercal y$.



      How to prove this assertion if valid? I haven't find out a clear path.










      share|cite|improve this question













      Suppose vectors $v_i, i=1, ..., p$ is a basis of the null space of $mtimes n$ matrix $B$. If $xin mathbbR^n, <x, v_i>=0, i=1, ..., p$, then there exists a vector $y$ such that



      $x=B^intercal y$.



      How to prove this assertion if valid? I haven't find out a clear path.







      linear-algebra






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Sep 11 at 2:45









      John Smith

      545




      545




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote



          accepted










          I leave you to prove that the null space of any matrix $A$ is the orthogonal complement of the column space of $A^T$.



          Using thay property, proving your assertion is straightforward.






          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: 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%2f2912682%2fnull-space-and-related-question%23new-answer', 'question_page');

            );

            Post as a guest






























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            1
            down vote



            accepted










            I leave you to prove that the null space of any matrix $A$ is the orthogonal complement of the column space of $A^T$.



            Using thay property, proving your assertion is straightforward.






            share|cite|improve this answer
























              up vote
              1
              down vote



              accepted










              I leave you to prove that the null space of any matrix $A$ is the orthogonal complement of the column space of $A^T$.



              Using thay property, proving your assertion is straightforward.






              share|cite|improve this answer






















                up vote
                1
                down vote



                accepted







                up vote
                1
                down vote



                accepted






                I leave you to prove that the null space of any matrix $A$ is the orthogonal complement of the column space of $A^T$.



                Using thay property, proving your assertion is straightforward.






                share|cite|improve this answer












                I leave you to prove that the null space of any matrix $A$ is the orthogonal complement of the column space of $A^T$.



                Using thay property, proving your assertion is straightforward.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Sep 11 at 3:06









                Javi

                3349




                3349



























                     

                    draft saved


                    draft discarded















































                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2912682%2fnull-space-and-related-question%23new-answer', 'question_page');

                    );

                    Post as a guest













































































                    這個網誌中的熱門文章

                    How to combine Bézier curves to a surface?

                    Mutual Information Always Non-negative

                    Why am i infinitely getting the same tweet with the Twitter Search API?