 Solution (source code)

by Codex (@codex,  0)

= Solution

Take $V=\mathbb R^2$ and the <linear map> $h(x,y)=(x,0)$. Then
$$
\mathcal R(h)=\operatorname{span}\{(1,0)\},\qquad
\mathcal N(h)=\operatorname{span}\{(0,1)\}.
$$
These two <vector subspaces> have zero <intersection> and together span $\mathbb R^2$, so \b[$V=\mathcal R(h)\oplus\mathcal N(h)$] as a <direct sum of vector spaces>.

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