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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).

mtheorylord
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    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 '18 at 22:46
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    I know no book with that statement: it follows directly from many other things. – DonAntonio Nov 14 '18 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 '18 at 23:16
  • I know I can show it myself but I am looking specifically for a book that says something like this. – mtheorylord Nov 14 '18 at 23:25
  • @JMoravitz It sounds strange, but I need a book that says something along these lines succintly. – mtheorylord Nov 14 '18 at 23:29
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    Why do you specifically need a book that explicitly says it? Certainly noone needs a citation to a book that explicitly says that $101\times 99 = 9999$ 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 '18 at 23:36
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    @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 '18 at 23:53
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    Concerning "In that form or something close to it": There are about 100 books fulfilling this condition interpreted as "close". Therefore you should state more precisely how close. Before that, you should reread your sentence and convert it to a semantically correct English sentence. – Christian Blatter Nov 17 '18 at 18:59

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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.

Mostafa Ayaz
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