Is corollary status warranted for this result?

[1MileCrash]

Is this really worthy of being pointed out as a corollary??

 

[phosgene]

Oops. It's just the contrapositive. Then no, it's not worthy of being a corollary.

 

[fluidistic]

If it follows from the theorem 1.6 then why not? :) Not really funny to me.

 

[1MileCrash]

Yes. Not all theorems work both ways like that.

Are you sure?

 

[phosgene]

Are you sure?

I misread it, thought that the corollary was about dependence too.

 

[1MileCrash]

I misread it, thought that the corollary was about dependence too.

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!)

 

[Office_Shredder]

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.

 

[1MileCrash]

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.

 

[AlephZero]

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.

 

[FlexGunship]

Is this really worthy of being pointed out as a corollary??

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