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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.
Linear subspace A linear subspace or vector subspace W of a vector space V is a non-empty subset of V that is closed under vector addition and scalar multiplication; that is, the sum of two elements of W and the product of an element of W by a scalar belong to W. [10] This implies that every linear combination of elements of W belongs to W. A ...
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
A subalgebra of an algebra over a commutative ring or field is a vector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field.
The set of tempered distributions forms a vector subspace of the space of distributions ′ and is thus one example of a space of distributions; there are many other spaces of distributions. There also exist other major classes of test functions that are not subsets of C c ∞ ( U ) , {\displaystyle C_{c}^{\infty }(U),} such as spaces of ...
The vector subspace is complemented in if and only if any of the following holds: [1] There exists a continuous linear map : with image = such that =. That is, is a ...
The sum of a closed vector subspace and a finite-dimensional vector subspace is closed. [6] If M {\displaystyle M} is a vector subspace of X {\displaystyle X} and N {\displaystyle N} is a closed neighborhood of the origin in X {\displaystyle X} such that U ∩ N {\displaystyle U\cap N} is closed in X {\displaystyle X} then M {\displaystyle M ...
Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one. The zero vector space is conceptually different from the null space of a linear operator L, which is the kernel of L.