Am I the only one that finds this funny?

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Discussion Overview

The discussion revolves around the classification of a mathematical statement as a corollary or a contrapositive, particularly in the context of linear dependence and independence in vector spaces. Participants explore the implications of these concepts and their relationships to theorems.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether a certain statement should be considered a corollary, with one suggesting it is merely a contrapositive.
  • Another participant argues that if a statement follows from a theorem, it could be deemed worthy of being a corollary.
  • There is a discussion about the implications of a corollary related to dependence, suggesting that if it were true, all subsets of a vector space would be linearly dependent.
  • One participant clarifies that every vector space contains the zero vector, making it linearly dependent, but removing it could lead to a linearly independent set.
  • Another participant emphasizes the importance of proving results from first principles to understand linear dependence and independence better.
  • There is a humorous exchange regarding the interpretation of a statement, with participants expressing confusion over its wording.

Areas of Agreement / Disagreement

Participants express differing views on whether the statement in question qualifies as a corollary or merely a contrapositive. The implications of linear dependence and independence also generate varied interpretations, indicating that the discussion remains unresolved.

Contextual Notes

Some assumptions about the definitions of corollaries and contrapositives are not explicitly stated, and the discussion includes unresolved mathematical implications regarding linear dependence in vector spaces.

1MileCrash
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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!)
 
Last edited:
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.
 
  • #10
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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