Am I the only one that finds this funny?

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The discussion revolves around the classification of a mathematical statement as a corollary versus a contrapositive. Participants debate whether a certain theorem's implications warrant being labeled as a corollary. Misunderstandings arise regarding the nature of dependence in vector spaces, with one participant suggesting that if a corollary were about dependence, it would imply that all subsets of a vector space are linearly dependent. This leads to clarification that while every vector space contains the zero vector, making it linearly dependent, it is possible to have linearly independent sets by removing certain elements. The conversation highlights the importance of understanding linear dependence and independence in mathematical proofs.
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Is this really worthy of being pointed out as a corollary??
 

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Oops. It's just the contrapositive. Then no, it's not worthy of being a corollary.
 
If it follows from the theorem 1.6 then why not? :) Not really funny to me.
 
phosgene said:
Yes. Not all theorems work both ways like that.

Are you sure? :wink:
 
1MileCrash said:
Are you sure? :wink:

I misread it, thought that the corollary was about dependence too. :redface:
 
phosgene said:
I misread it, thought that the corollary was about dependence too. :redface:

Hah, I don't blame you. I had to do a double (or triple) take.

Though, interestingly enough, if the corollary was about dependence, then that would mean (S1 dependent) <=> (S2 dependent), which would mean that any subset or superset of any linearly dependent set would be linearly dependent. Which would mean that all subsets of a vector space would be linearly dependent, because all vector spaces must contain some linearly dependent subset (namely itself) (unless I'm just confusing myself here!)
 
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That's true, but not for the reason that you seem to be thinking. Every vector space contains 0, and therefore is a linearly dependent set, but if you remove that point you could have a linearly independent set. For example (and probably the only example), F2 the field of two elements, as a one dimensional vector space over itself.
 
Office_Shredder said:
That's true, but not for the reason that you seem to be thinking. Every vector space contains 0, and therefore is a linearly dependent set, but if you remove that point you could have a linearly independent set. For example (and probably the only example), F2 the field of two elements, as a one dimensional vector space over itself.

I'm saying - if the corollary was talking about dependence, a consequence would be that any subset of a vector space is linearly dependent because the vector space itself is.
 
I think it's not just to "point it out", but ask you to prove both results from first principles, to get used to making proofs about linear dependence and independence.
 
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1MileCrash said:
Is this really worthy of being pointed out as a corollary??

Wait. Why? Because there's two extra commas in there?
 
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