Find a concept
Start typing to search the mathematical index.
If codim(L)=1 and x0∉L, then X=L⊕span{x0}.
Let be a vector space and let be a subspace with codim(L) . If , then:
Proposition:
where 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.
A vector space over a field is a set with operations (functions) and , and an element , satisfying the following for all and :
Let be a vector space over a field , and let . The set is a linear subspace of if
With the inherited operations, is itself a vector space.
Let be a vector space and let be a linear subspace. The codimension of in is
where is the quotient space of by .
Codimension measures "how many independent linear constraints" define . Codimension one subspaces play a special role in the geometry of hyperplanes.
Let be a vector space and let be linear subspaces.
Let be their (set) sum. We say that is the direct sum of and if
In this case we write
Direct sums formalize "adding independent directions": if , then no nonzero vector lies in both subspaces. The key structural property is uniqueness of decomposition, captured in the direct sum characterization.