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











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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







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  • 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
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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.






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






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      2 Answers
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      2 Answers
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      up vote
      0
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      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






















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






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






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






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







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                  answered 37 mins ago









                  multicusp

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