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Ω is affine iff Ω−ω is a linear subspace (equivalently, Ω=ω+L).
Let be a vector space and let be nonempty.
Lemma: The set is affine if and only if for every , the translate
is a linear subspace of .
Equivalently, is affine iff there exist and a subspace such that .
Context: This lemma explains why affine sets are often called "affine subspaces": they are precisely translates of linear subspaces.
A vector space over a field is a set equipped with two operations (functions) and such that for all and :
Vector spaces are the basic objects studied via linear maps and invariants such as the determinant and eigenvalues of operators.
Let be a vector space over , and let .
The set is a linear subspace of if:
With the inherited operations, is itself a vector space, and many constructions in analysis arise as subspaces (kernels, images, function spaces, etc.). Subspaces are also the building blocks for quotient spaces and direct sums.