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!
I'll assume you know how to prove that if $\displaystyle U+W=V$ then $\displaystyle U\neq W$ (it is trivial).
Assume U,W are subspaces of V with $\displaystyle dim V = n, ~ dimU = dim W = n-1$ such that $\displaystyle U \neq W$.
Let $\displaystyle B = \{w_1,w_2,...,w_{n-1}\}$ be a basis for W.
Let $\displaystyle u \in U - W ~ \text{s.t.} ~ u \neq 0$ (we know one such element exists otherwise $\displaystyle U=W$ since $\displaystyle 0\in U, ~ 0 \in W$).
Then u is not spanned by $\displaystyle \{w_1,w_2,...,w_{n-1}\}$, otherwise it would be an element of W (by definition). Then, $\displaystyle B' = \{w_1,w_2,...,w_{n-1},u\}$ is an independent set and obviously $\displaystyle dim B' = n$ and therefore it is a basis of V.
This gives us that $\displaystyle V \subset U+W$.
However, obviously there can be no element in $\displaystyle U+W$ that is not in V since they are subspaces, therefore $\displaystyle U+W \subset V \rightarrow U+W = V$
By the way, what uni are you studying in?