I'm new to this board and although you guys here have helped me a lot in the past, this is my first time to ask a question here. I couldn't find anything similar so far and I'd be grateful for any help. The following is a "Prove or disprove" type of problem:
Prove or disprove: Let $\mathscr{U,V}\subseteq\mathbb{R}^n$ be two subspaces with $\mathscr{U}\cap\mathscr{V}=\{0\}$ and $\mathscr{U}\oplus\mathscr{V}=\mathbb{R}^n$. Let $\mathbf{u}\in\mathscr{U}$ and $\mathbf{v}\in\mathscr{V}$. Then $\langle\mathbf{u,v}\rangle=0$.
Now I usuallly don't have problems to at least find a starting point for proofs like this, but I'm not sure with this one. Now from the properties of the two subspaces I would have concluded that they form a linearly independent basis for $\mathbb{R}^n$. Therefore, the inner product can be zero iff either $\mathbf{u}=0$ or $\mathbf{v}=0$ or if they are orthogonal to each other. But I honestly have no idea where this would lead me.
Any suggestions on how to start this problem? Maybe I am just overlooking something pretty obvious, since it seems like it shouldn't be too hard to find an answer once you know where to begin. Thanks in advance (and also sorry for the poor formatting in the title).
Hint: What is the standard definition of $\langle,\rangle$? Try to find two vectors that span $R^2$ but are not orthogonal. Generalize to $R^n$.