Search results
Results from the WOW.Com Content Network
In the language of universal algebra, a vector space is an algebra over the universal vector space R ∞ of finite sequences of coefficients, corresponding to finite sums of vectors, while an affine space is an algebra over the universal affine hyperplane in this space (of finite sequences summing to 1), a cone is an algebra over the universal ...
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.
When the scalar field is the real numbers, the vector space is called a real vector space, and when the scalar field is the complex numbers, the vector space is called a complex vector space. [4] These two cases are the most common ones, but vector spaces with scalars in an arbitrary field F are also commonly considered.
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.
If is a subset of a vector space then a linear sub-variety (that is, an affine subspace) of the vector space is called a support variety if meets (that is, is not empty) and every open segment whose interior meets is necessarily a subset of . [3] A 0-dimensional support variety is called an extreme point of . [3]
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. [1] A norm is a generalization of the intuitive notion of "length" in the physical world.
In the case of a vector space or a free module, the gradation is the gradation of the polynomials by the total degree. A non-free module can be written as L / M , where L is a free module of base B ; its symmetric algebra is the quotient of the (graded) symmetric algebra of L (a polynomial ring) by the homogeneous ideal generated by the ...
A topological vector space homomorphism (abbreviated TVS homomorphism), also called a topological homomorphism, [2] [3] is a continuous linear map: between topological vector spaces (TVSs) such that the induced map : is an open mapping when := (), which is the range or image of , is given the subspace topology induced by .