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If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations of V.Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w 1, w 2 are elements of W and α, β are elements of K, it follows that αw 1 + βw 2 is in W.
A linear subspace is a vector space for the induced addition and scalar multiplication; this means that the closure property implies that the axioms of a vector space are satisfied. [11] The closure property also implies that every intersection of linear subspaces is a linear subspace. [11] Linear span
Linear subspace, in linear algebra, a subset of a vector space that is closed under addition and scalar multiplication; Flat (geometry), a Euclidean subspace; Affine subspace, a geometric structure that generalizes the affine properties of a flat; Projective subspace, a geometric structure that generalizes a linear subspace of a vector space
Given a vector space V over a field K, the span of a set S of vectors (not necessarily finite) is defined to be the intersection W of all subspaces of V that contain S. It is thus the smallest (for set inclusion) subspace containing W. It is referred to as the subspace spanned by S, or by the vectors in S.
If is a vector subspace of a Hilbert space the orthogonal complement of the orthogonal complement of is the closure of , that is, = ¯. Some other useful properties that always hold are the following.
The space of (linear) complementary subspaces of a vector subspace V in a vector space W is an affine space, over Hom(W/V, V). That is, if 0 → V → W → X → 0 is a short exact sequence of vector spaces, then the space of all splittings of the exact sequence naturally carries the structure of an affine space over Hom( X , V ) .
A subspace is Lagrangian if and only if it is both isotropic and coisotropic. In a finite-dimensional vector space, a Lagrangian subspace is an isotropic one whose dimension is half that of V. Every isotropic subspace can be extended to a Lagrangian one. Referring to the canonical vector space R 2n above, the subspace spanned by {x 1, y 1} is ...
Formally, the construction is as follows. [1] Let be a vector space over a field, and let be a subspace of .We define an equivalence relation on by stating that iff .That is, is related to if and only if one can be obtained from the other by adding an element of .