May 26th, 2016, 07:35 AM  #11  
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  Quote:
 
May 26th, 2016, 08:14 AM  #12 
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271 
qwerty you are quite correct to point out, as Archie has already done, that the sets in vector spaces include a uniqueness requirement. Zylo, of all people, should know this since he only accepts the Zermelo set definition. The real purpose of the wiki definition is the following situation for three variables w,x, y w + x = 4 y  x = 6 wy = 10 these are linearly dependent although each only contains two of the three variables. Any two are independent. Take the first two, you need a third equation to pick out the third variable. eg combining the first two gives you an equation between w and y since you can eliminate x. but the third equation will not do since it is simply the linear combination multiplied by 1. Note the key phrase S is a subset, not necessarily the entire set. Last edited by studiot; May 26th, 2016 at 08:25 AM. 
May 26th, 2016, 09:02 AM  #13  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
Technically yes. That's why "collection" is often used. There is no problem if you want to discuss a general set of vectors S=(X1, X2,... ). where Xi=Xj is a possibility. You could assume this possibility is ruled out by definition of a set, in which case the wiki definition is redundant. But I think that's just a sophistry. You are not thinking about set theory in elementary linear algebra. So just think about the wiki definition as ruling out equal vectors ruining an otherwise LI set. The whole thing becomes foggy if you have X and 2X in the set. Personally, I would use the std def where the type of question raised here is automatically taken care of by context in a particular case: 2 vectors equal? The "set" is LD. But let's check the LD by not considering one of the equal vectors. X1=2(X2)? Sane thing. Set is LD but you might want to check it without X.  
May 26th, 2016, 09:28 AM  #14  
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271  Quote:
Infinite sets in linear algebra can lead to some unexpected consequences, for example the linear decomposition of a square wave in fourier series. This is a good reason to stick with the type of vectors I was offering rather than the most general vector space.  
May 27th, 2016, 02:05 AM  #15 
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271 
qwerty, do you understand what sort of vector w, x and y are in my example in post 12? They are not geometric or row or column or any sort of ntuple. They are algebraic variables. That is maps or functions from R to R The rest of the numbers are scalar constants, also from from R This is an example of what you were asking in your post7 
May 27th, 2016, 04:26 AM  #16  
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  Quote:
Last edited by 123qwerty; May 27th, 2016 at 04:35 AM.  

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