Proving $V / U \cong W$ for complementary subspaces $U, W$ without using the dimension formula

Question

Let $V$ be a finite-dimensional vector space and $U,W \subseteq V$ be disjoint subspaces of $V$ such that $U+W=V$. How can it be shown that $V / U \cong W$ without using the dimension formula(e)? I've seen that fact used in a equivalence proof between the different dimension formulae, but I don't know how to prove it without proving the dimension formulae first. I guess it comes down to finding an explicit basis of $V / U$ but I don't quite know how to do that.

Answer 1

The way is proving that the restriction to $W$ of the quotient projection is injective and surjective (onto $V/U$). It's really just some definition-fu, where you use $U\cap W=\{0\}$ and $U+W=V$ respectively.

Answer 2

Hint:

Consider the map: \begin{align} p\colon V&\longrightarrow W,\\ v&\longmapsto w\enspace\text{if }\; v=u+w. \end{align} Show $p$ is well-defined, and it is linear surjective. What is its kernel?