If V and W are linear subspaces of $R^n$, then V transversal W means just $V + W = R^n$

Question

If V and W are linear subspaces of $R^n$, then V transversal W means just $V + W = R^n$.

Could anyone give me a hint for this exercise please?

Answer 1

So to answer this question, you have to think of $V$ and $W$ as submanifolds of $\mathbb{R}^n$. They intersect transversally if, at every point of their intersection, the sum of their tangent spaces is $\mathbb{R}^n$. But remember that since $V$ and $W$ are vector spaces, their tangent spaces at any point are isomorphically just themselves, and we identify them as such. The same goes for the tangent space to $\mathbb{R}^n$ at any point. Since all these identifications commute in a sense that can be made precise, there is no conflict if we make all these identifications simultaneously and then think about just the subspaces themselves. Thinking about what this means in the context of transversality, we realize that this is just the condition that $V+W = \mathbb{R}^n$.