Books that state column vectors are linearly dependent if determinant is $0$?











up vote
0
down vote

favorite
1












I am looking for some linear algebra books that contain the below proposition:



If a matrix over an arbitrary vector space $V$ and base field $mathbb{F}$ that has determinant $0$ if the columns are linearly dependent.



In that form or something close to it. By this, I mean that the book actually states the above in a proposition or maybe as a combination of $2$ side by side lemmas (without the reader required to his own logic).



Edit:
I know how to prove this fact, and where the proof comes from, etc. but, long story short, I just need to find some book that states something similar to this succinctly (even without proof is fine).






reference-request book-recommendation







share|cite|improve this question

















This question has an open bounty worth +50
reputation from mtheorylord ending in 12 hours.


The current answers do not contain enough detail.












  • 1




    You are missing a number of key words such as how the matrix must be square and you use 'if' when you should have used 'iff'... but yes, this is one of the many equivalent statements in the invertible matrix theorem. Just about every introductory linear algebra textbook (with at least some proof writing component) will have a proof for the result where the field is real numbers or has this as an exercise for the reader, and can be sufficiently generalized to work as a proof for arbitrary vector spaces and fields.
    – JMoravitz
    Nov 14 at 22:46








  • 1




    I know no book with that statement: it follows directly from many other things.
    – DonAntonio
    Nov 14 at 22:47










  • You can show it yourself, assuming some other known fact. For example, for square $A $, the determinant is non-zero iff $A $ is invertible. This happens iff $Ax=0$ has only the trivial solution $x=0$. The columns are lin indep iff the only solution to the above equation is the trivial solution (Check!)
    – AnyAD
    Nov 14 at 23:16








  • 2




    Why do you specifically need a book that explicitly says it? Certainly noone needs a citation to a book that explicitly says that $101times 99 = 9999$ to be able to use the information in a report... you can show it yourself or allude to the fact that every reader with the expected amount of prerequisite skill to be reading your paper should be able to show it themselves. It shouldn't be explicitly necessary here either.
    – JMoravitz
    Nov 14 at 23:36






  • 1




    @JMoravitz I made a sort of bet, it's a pretty long story. I let you guys in on it if I can find this fact somewhere.
    – mtheorylord
    Nov 14 at 23:53

















up vote
0
down vote

favorite
1












I am looking for some linear algebra books that contain the below proposition:



If a matrix over an arbitrary vector space $V$ and base field $mathbb{F}$ that has determinant $0$ if the columns are linearly dependent.



In that form or something close to it. By this, I mean that the book actually states the above in a proposition or maybe as a combination of $2$ side by side lemmas (without the reader required to his own logic).



Edit:
I know how to prove this fact, and where the proof comes from, etc. but, long story short, I just need to find some book that states something similar to this succinctly (even without proof is fine).






reference-request book-recommendation







share|cite|improve this question

















This question has an open bounty worth +50
reputation from mtheorylord ending in 12 hours.


The current answers do not contain enough detail.












  • 1




    You are missing a number of key words such as how the matrix must be square and you use 'if' when you should have used 'iff'... but yes, this is one of the many equivalent statements in the invertible matrix theorem. Just about every introductory linear algebra textbook (with at least some proof writing component) will have a proof for the result where the field is real numbers or has this as an exercise for the reader, and can be sufficiently generalized to work as a proof for arbitrary vector spaces and fields.
    – JMoravitz
    Nov 14 at 22:46








  • 1




    I know no book with that statement: it follows directly from many other things.
    – DonAntonio
    Nov 14 at 22:47










  • You can show it yourself, assuming some other known fact. For example, for square $A $, the determinant is non-zero iff $A $ is invertible. This happens iff $Ax=0$ has only the trivial solution $x=0$. The columns are lin indep iff the only solution to the above equation is the trivial solution (Check!)
    – AnyAD
    Nov 14 at 23:16








  • 2




    Why do you specifically need a book that explicitly says it? Certainly noone needs a citation to a book that explicitly says that $101times 99 = 9999$ to be able to use the information in a report... you can show it yourself or allude to the fact that every reader with the expected amount of prerequisite skill to be reading your paper should be able to show it themselves. It shouldn't be explicitly necessary here either.
    – JMoravitz
    Nov 14 at 23:36






  • 1




    @JMoravitz I made a sort of bet, it's a pretty long story. I let you guys in on it if I can find this fact somewhere.
    – mtheorylord
    Nov 14 at 23:53















up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





I am looking for some linear algebra books that contain the below proposition:



If a matrix over an arbitrary vector space $V$ and base field $mathbb{F}$ that has determinant $0$ if the columns are linearly dependent.



In that form or something close to it. By this, I mean that the book actually states the above in a proposition or maybe as a combination of $2$ side by side lemmas (without the reader required to his own logic).



Edit:
I know how to prove this fact, and where the proof comes from, etc. but, long story short, I just need to find some book that states something similar to this succinctly (even without proof is fine).






reference-request book-recommendation







share|cite|improve this question















I am looking for some linear algebra books that contain the below proposition:



If a matrix over an arbitrary vector space $V$ and base field $mathbb{F}$ that has determinant $0$ if the columns are linearly dependent.



In that form or something close to it. By this, I mean that the book actually states the above in a proposition or maybe as a combination of $2$ side by side lemmas (without the reader required to his own logic).



Edit:
I know how to prove this fact, and where the proof comes from, etc. but, long story short, I just need to find some book that states something similar to this succinctly (even without proof is fine).






reference-request book-recommendation




reference-request book-recommendation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 17 at 19:07

























asked Nov 14 at 22:38









mtheorylord

1,751625




1,751625






This question has an open bounty worth +50
reputation from mtheorylord ending in 12 hours.


The current answers do not contain enough detail.








This question has an open bounty worth +50
reputation from mtheorylord ending in 12 hours.


The current answers do not contain enough detail.










  • 1




    You are missing a number of key words such as how the matrix must be square and you use 'if' when you should have used 'iff'... but yes, this is one of the many equivalent statements in the invertible matrix theorem. Just about every introductory linear algebra textbook (with at least some proof writing component) will have a proof for the result where the field is real numbers or has this as an exercise for the reader, and can be sufficiently generalized to work as a proof for arbitrary vector spaces and fields.
    – JMoravitz
    Nov 14 at 22:46








  • 1




    I know no book with that statement: it follows directly from many other things.
    – DonAntonio
    Nov 14 at 22:47










  • You can show it yourself, assuming some other known fact. For example, for square $A $, the determinant is non-zero iff $A $ is invertible. This happens iff $Ax=0$ has only the trivial solution $x=0$. The columns are lin indep iff the only solution to the above equation is the trivial solution (Check!)
    – AnyAD
    Nov 14 at 23:16








  • 2




    Why do you specifically need a book that explicitly says it? Certainly noone needs a citation to a book that explicitly says that $101times 99 = 9999$ to be able to use the information in a report... you can show it yourself or allude to the fact that every reader with the expected amount of prerequisite skill to be reading your paper should be able to show it themselves. It shouldn't be explicitly necessary here either.
    – JMoravitz
    Nov 14 at 23:36






  • 1




    @JMoravitz I made a sort of bet, it's a pretty long story. I let you guys in on it if I can find this fact somewhere.
    – mtheorylord
    Nov 14 at 23:53
















  • 1




    You are missing a number of key words such as how the matrix must be square and you use 'if' when you should have used 'iff'... but yes, this is one of the many equivalent statements in the invertible matrix theorem. Just about every introductory linear algebra textbook (with at least some proof writing component) will have a proof for the result where the field is real numbers or has this as an exercise for the reader, and can be sufficiently generalized to work as a proof for arbitrary vector spaces and fields.
    – JMoravitz
    Nov 14 at 22:46








  • 1




    I know no book with that statement: it follows directly from many other things.
    – DonAntonio
    Nov 14 at 22:47










  • You can show it yourself, assuming some other known fact. For example, for square $A $, the determinant is non-zero iff $A $ is invertible. This happens iff $Ax=0$ has only the trivial solution $x=0$. The columns are lin indep iff the only solution to the above equation is the trivial solution (Check!)
    – AnyAD
    Nov 14 at 23:16








  • 2




    Why do you specifically need a book that explicitly says it? Certainly noone needs a citation to a book that explicitly says that $101times 99 = 9999$ to be able to use the information in a report... you can show it yourself or allude to the fact that every reader with the expected amount of prerequisite skill to be reading your paper should be able to show it themselves. It shouldn't be explicitly necessary here either.
    – JMoravitz
    Nov 14 at 23:36






  • 1




    @JMoravitz I made a sort of bet, it's a pretty long story. I let you guys in on it if I can find this fact somewhere.
    – mtheorylord
    Nov 14 at 23:53










1




1




You are missing a number of key words such as how the matrix must be square and you use 'if' when you should have used 'iff'... but yes, this is one of the many equivalent statements in the invertible matrix theorem. Just about every introductory linear algebra textbook (with at least some proof writing component) will have a proof for the result where the field is real numbers or has this as an exercise for the reader, and can be sufficiently generalized to work as a proof for arbitrary vector spaces and fields.
– JMoravitz
Nov 14 at 22:46






You are missing a number of key words such as how the matrix must be square and you use 'if' when you should have used 'iff'... but yes, this is one of the many equivalent statements in the invertible matrix theorem. Just about every introductory linear algebra textbook (with at least some proof writing component) will have a proof for the result where the field is real numbers or has this as an exercise for the reader, and can be sufficiently generalized to work as a proof for arbitrary vector spaces and fields.
– JMoravitz
Nov 14 at 22:46






1




1




I know no book with that statement: it follows directly from many other things.
– DonAntonio
Nov 14 at 22:47




I know no book with that statement: it follows directly from many other things.
– DonAntonio
Nov 14 at 22:47












You can show it yourself, assuming some other known fact. For example, for square $A $, the determinant is non-zero iff $A $ is invertible. This happens iff $Ax=0$ has only the trivial solution $x=0$. The columns are lin indep iff the only solution to the above equation is the trivial solution (Check!)
– AnyAD
Nov 14 at 23:16






You can show it yourself, assuming some other known fact. For example, for square $A $, the determinant is non-zero iff $A $ is invertible. This happens iff $Ax=0$ has only the trivial solution $x=0$. The columns are lin indep iff the only solution to the above equation is the trivial solution (Check!)
– AnyAD
Nov 14 at 23:16






2




2




Why do you specifically need a book that explicitly says it? Certainly noone needs a citation to a book that explicitly says that $101times 99 = 9999$ to be able to use the information in a report... you can show it yourself or allude to the fact that every reader with the expected amount of prerequisite skill to be reading your paper should be able to show it themselves. It shouldn't be explicitly necessary here either.
– JMoravitz
Nov 14 at 23:36




Why do you specifically need a book that explicitly says it? Certainly noone needs a citation to a book that explicitly says that $101times 99 = 9999$ to be able to use the information in a report... you can show it yourself or allude to the fact that every reader with the expected amount of prerequisite skill to be reading your paper should be able to show it themselves. It shouldn't be explicitly necessary here either.
– JMoravitz
Nov 14 at 23:36




1




1




@JMoravitz I made a sort of bet, it's a pretty long story. I let you guys in on it if I can find this fact somewhere.
– mtheorylord
Nov 14 at 23:53






@JMoravitz I made a sort of bet, it's a pretty long story. I let you guys in on it if I can find this fact somewhere.
– mtheorylord
Nov 14 at 23:53












2 Answers
2






active

oldest

votes

















up vote
0
down vote













Does the following book help you?



https://www.amazon.com/Linear-Algebra-2nd-Kenneth-Hoffman/dp/0135367972



Please inform me whether it was helpful or I should delete the answer.






share|cite|improve this answer




























    up vote
    0
    down vote













    Linear Algebra 1, Martin Otto, Winter Term 2013/14, Definition 2.4.5






    share|cite|improve this answer








    New contributor




    multicusp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.


















      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%2f2998919%2fbooks-that-state-column-vectors-are-linearly-dependent-if-determinant-is-0%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








      up vote
      0
      down vote













      Does the following book help you?



      https://www.amazon.com/Linear-Algebra-2nd-Kenneth-Hoffman/dp/0135367972



      Please inform me whether it was helpful or I should delete the answer.






      share|cite|improve this answer

























        up vote
        0
        down vote













        Does the following book help you?



        https://www.amazon.com/Linear-Algebra-2nd-Kenneth-Hoffman/dp/0135367972



        Please inform me whether it was helpful or I should delete the answer.






        share|cite|improve this answer























          up vote
          0
          down vote










          up vote
          0
          down vote









          Does the following book help you?



          https://www.amazon.com/Linear-Algebra-2nd-Kenneth-Hoffman/dp/0135367972



          Please inform me whether it was helpful or I should delete the answer.






          share|cite|improve this answer












          Does the following book help you?



          https://www.amazon.com/Linear-Algebra-2nd-Kenneth-Hoffman/dp/0135367972



          Please inform me whether it was helpful or I should delete the answer.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 2 days ago









          Mostafa Ayaz

          12k3733




          12k3733






















              up vote
              0
              down vote













              Linear Algebra 1, Martin Otto, Winter Term 2013/14, Definition 2.4.5






              share|cite|improve this answer








              New contributor




              multicusp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.






















                up vote
                0
                down vote













                Linear Algebra 1, Martin Otto, Winter Term 2013/14, Definition 2.4.5






                share|cite|improve this answer








                New contributor




                multicusp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.




















                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  Linear Algebra 1, Martin Otto, Winter Term 2013/14, Definition 2.4.5






                  share|cite|improve this answer








                  New contributor




                  multicusp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.









                  Linear Algebra 1, Martin Otto, Winter Term 2013/14, Definition 2.4.5







                  share|cite|improve this answer








                  New contributor




                  multicusp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.









                  share|cite|improve this answer



                  share|cite|improve this answer






                  New contributor




                  multicusp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.









                  answered 37 mins ago









                  multicusp

                  1




                  1




                  New contributor




                  multicusp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.





                  New contributor





                  multicusp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.






                  multicusp is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.






























                       

                      draft saved


                      draft discarded



















































                       


                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2998919%2fbooks-that-state-column-vectors-are-linearly-dependent-if-determinant-is-0%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

                      Probability when a professor distributes a quiz and homework assignment to a class of n students.

                      Image
                      0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
                      Read more

                      Aardman Animations

                      Image
                      Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
                      Read more

                      Are they similar matrix

                      Image
                      0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
                      Read more