# Why does the union of $k$-dimensional subspaces contain no „new“ $k$-dimensional subspaces?

For simplification, let $$V$$ be a finite-dimensional, real vector space. I know that I cannot represent $$V$$ as the union of finitely many proper subspaces (or even just $$k-1$$-dimensional subspaces). But how about the following:

Let $$W_1,…,W_n\leq V$$ be $$k$$-dimensional subspaces. Then every $$k$$-dimensional subspace of $$W_1\cup \ldots \cup W_n$$ must coincide with one of the $$W_i$$.

My initial idea was to choose a basis and trace where in which $$W_i$$ they could possibly „land“ on the right hand side, but I didn't find a proof based on this.

Two non-equal $$k$$-dimensional subspaces must intersect in a $$k-1$$-dimensional subspace. If some new subspace were to exist, it would have to be the union of $$k$$ $$k-1$$ dimensional subspaces, which as you have correctly stated is impossible.