Linear Algebra Done Right · Problem 1C.22

The key is recognizing that $U$ is a 2-dimensional subspace (parametrized by the free variables $x$ and $y$), so you need to find three complementary subspaces whose dimensions sum with $\dim(U) = 2$ to give $5$. Start by writing vectors in $U$ as linear combinations of basis vectors, then identify which coordinate directions are "used up" by $U$ and which are "free"—those free directions will span your complementary subspaces. The direct sum decomposition works when the subspaces are independent and their dimensions add to $5$.

The key is recognizing that $U$ is a 2-dimensional subspace (parametrized by the free variables $x$ and $y$), so you need to find three complementary subspaces whose dimensions sum with $\dim(U) = 2$ to give $5$. Start by writing vectors in $U$ as linear combinations of basis vectors, then identify which coordinate directions are "used up" by $U$ and which are "free"—those free directions will span your complementary subspaces. The direct sum decomposition works when the subspaces are independent and their dimensions add to $5$.