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April 5th, 2013, 11:43 AM  #1 
Senior Member Joined: Aug 2011 Posts: 149 Thanks: 0  Need help with proving vector space over corpus
Hello. Could anyone give me some hints, ideas or suggestions on solving the following: Let V be n dimensional vector space over corpus K. Suppose we have r dimensional subspace where r < n. Prove that W = Y where It would be great if I could also get some explanations for more complex things abut why something works as it works (in order to understand it better). Or if someone has seen proof to this on some book, link would be great as well. 
April 5th, 2013, 02:10 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,436 Thanks: 562  Re: Need help with proving vector space over corpus
The definition of Y needs clarification.

April 6th, 2013, 05:23 AM  #3 
Senior Member Joined: Aug 2011 Posts: 149 Thanks: 0  Re: Need help with proving vector space over corpus
doesnt this define Y? I am not sure how to understand this \bigcap and dimU. This task has also hint: To prove that W ? Y we show that existence of v ? Y\W leads to conflict. 
April 6th, 2013, 01:17 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,436 Thanks: 562  Re: Need help with proving vector space over corpus
The notation of the symbols between { and } needs explanation, particularly between : and last ,.

April 6th, 2013, 01:20 PM  #5 
Senior Member Joined: Aug 2011 Posts: 149 Thanks: 0  Re: Need help with proving vector space over corpus
Y should be subspace

April 8th, 2013, 07:54 AM  #6 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Need help with proving vector space over corpus
You appear to be saying that Y is the intersection of all n1 dimensional subspaces of V. But I don't know what that last is supposed to mean. It is, however, easy to show that the intersection of all n1 dimensional subspaces of V is just the 0 vector. 
April 11th, 2013, 10:12 AM  #7  
Senior Member Joined: Aug 2011 Posts: 149 Thanks: 0  Re: Need help with proving vector space over corpus Quote:
 
April 11th, 2013, 10:53 AM  #8 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Need help with proving vector space over corpus
Let v be any nonzero vector in the space. There exist a basis for the vector space that includes v. Let W be the subspace spanned by all vectors in that basis except v. Then W is an n1 dimensional subspace that does not include v.

April 11th, 2013, 11:46 AM  #9 
Senior Member Joined: Aug 2011 Posts: 149 Thanks: 0  Re: Need help with proving vector space over corpus
Oops, t looks like I've made typo. Itshould be: It should be W ? U instead Let V be n dimensional vector space over corpus K. Suppose we have r dimensional subspace where r < n. Prove that W = Y where It would be great if I could also get some explanations for more complex things abut why something works as it works (in order to understand it better). Or if someone has seen proof to this on some book, link would be great as well. 
April 13th, 2013, 06:56 AM  #10 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Need help with proving vector space over corpus
Since U is a subspace of V, it has a basis. Extend that basis to a basis for V. Remove a single vector from that basis, not one that is in the basis for U, to get an n1 dimensional subspace that contains U. The intersection of such subspaces contains all vectors in the basis for U but does not contain any basis vector that is not in that set.


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corpus, proving, space, vector 
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