Let V be vector space with dim(V)=n. And let U,W be subspaces with n-1 dimension each,I need to prove that: U+W=V if and only if U!=W

Thank you all!

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- December 2nd 2009, 12:05 PMAlso sprach ZarathustraSubspaces and Dimension
**Let V be vector space with dim(V)=n. And let U,W be subspaces with n-1 dimension each,I need to prove that****: U+W=V if and only if U!=W**

Thank you all! - December 2nd 2009, 01:10 PMDefunkt
I'll assume you know how to prove that if then (it is trivial).

Assume U,W are subspaces of V with such that .

Let be a basis for W.

Let (we know one such element exists otherwise since ).

Then u is not spanned by , otherwise it would be an element of W (by definition). Then, is an independent set and obviously and therefore it is a basis of V.

This gives us that .

However, obviously there can be no element in that is not in V since they are subspaces, therefore

By the way, what uni are you studying in? - December 2nd 2009, 01:22 PMAlso sprach Zarathustra
Thanks a lot! But, your assumption is wrong, can you prove the first line please?

- December 2nd 2009, 01:24 PMDefunkt
Well, assume . Assume, by contradiction, that . Then

What does this give us? (hint: look at the dimensions) - December 2nd 2009, 01:33 PMAlso sprach Zarathustra
That the dimension of U and W is not the dimension of V. Paradox?

I am really bad at it! Thank you for your patients! - December 2nd 2009, 01:37 PMDefunkt
Yes, that is correct! since , we must have that but in this case , therefore we reach a contradiction, and then .