Complement of subspace in terms of direct sum

Question

Let $U\subset W$ be a vector space. Show that there exists another subspace $V$ such that $W=U\bigoplus V$. What can be a candidate of such subspace?

Answer 1

Recall that a vector space over a field $K$ is a free, hence projective, $K$-module. Thus the exact sequence $$O\to U\hookrightarrow W\xrightarrow{\pi} W\diagup U\to O$$ splits, that's there exists a $K$-linear map $\varkappa:W\diagup U\to W$ such that $\pi\circ\varkappa = \mathrm{id}|W\diagup U$. The image $V=\mathrm{Im}(\varkappa)\subseteq W$ is the required subspace.