Codimension-One Subspaces Give Direct Sum Decompositions

If codim(L)=1 and x0∉L, then X=L⊕span{x0}.

Codimension-One Subspaces Give Direct Sum Decompositions

Let $X$ be a vector space and let $L\subset X$ be a subspace with codim(L) $=1$ . If $x_0\notin L$ , then:

Proposition:

X = L \oplus \operatorname{span}\{x_0\},

where $\oplus$ denotes the direct sum .

Context: This shows that a codimension-one subspace is “one linear dimension short” of the whole space. It is the structural fact behind representing hyperplanes as kernels (or level sets) of nonzero linear functionals.