Linear Subspaces, Transversal intersections I was told that if $V$ and $W$ are two subspaces of $\mathbb{R}^n$, then $V$ intersects W transversally means that $V+W = \mathbb{R}^n$.
My question is why? Is there a way to prove this? I feel like some type of inclusion/exclusion argument can be formed.

The definition of transversality for submanifolds $V$ and $W$ of a smooth manifold $S$ is that at every point where $V$ and $W$ intersect, the tangent space of $S$ is the span of the tangent spaces of $V$ and $W$.  In the case of linear subspaces of a linear space, the tangent spaces of $V$, $W$ and $S$ are naturally identified with $V$, $W$ and $S$ themselves.
