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 April 5th, 2013, 10:43 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 $W \cup V$ where r < n. Prove that W = Y where $Y= \bigcap \{U: U is V subspace, dimU = n -1, W \cup U\}$ 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, 01:10 PM #2 Global Moderator   Joined: May 2007 Posts: 6,555 Thanks: 599 Re: Need help with proving vector space over corpus The definition of Y needs clarification.
 April 6th, 2013, 04:23 AM #3 Senior Member   Joined: Aug 2011 Posts: 149 Thanks: 0 Re: Need help with proving vector space over corpus doesnt this define Y? $Y= \bigcap \{U: U is V subspace, dimU = n -1, W \cup U\}$ 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, 12:17 PM #4 Global Moderator   Joined: May 2007 Posts: 6,555 Thanks: 599 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, 12:20 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, 06:54 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 n-1 dimensional subspaces of V. But I don't know what that last $U\cup W$ is supposed to mean. It is, however, easy to show that the intersection of all n-1 dimensional subspaces of V is just the 0 vector.
April 11th, 2013, 09:12 AM   #7
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Re: Need help with proving vector space over corpus

Quote:
 Originally Posted by HallsofIvy You appear to be saying that Y is the intersection of all n-1 dimensional subspaces of V. But I don't know what that last $U\cup W$ is supposed to mean. It is, however, easy to show that the intersection of all n-1 dimensional subspaces of V is just the 0 vector.
How would you show that?

 April 11th, 2013, 09:53 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 non-zero 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 n-1 dimensional subspace that does not include v.
 April 11th, 2013, 10:46 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 $W \subset V$ where r < n. Prove that W = Y where $Y= \bigcap \{U: U is V subspace, dimU = n -1, W \subset U\}$ 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, 05:56 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 n-1 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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