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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.
This is an illustration of the closest vector problem (basis vectors in blue, external vector in green, closest vector in red). In CVP, a basis of a vector space V and a metric M (often L 2) are given for a lattice L, as well as a vector v in V but not necessarily in L. It is desired to find the vector in L closest to v (as measured by M).
In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic, more general (not limited to the finite-dimensional case), and conceptually simpler, although more abstract. A vector space over a field F (often the field of the real numbers) is a set V equipped with two binary operations.
A multi-objective optimization problem is a special case of a vector optimization problem: The objective space is the finite dimensional Euclidean space partially ordered by the component-wise "less than or equal to" ordering.
Every operator on a non-trivial complex finite dimensional vector space has an eigenvector, solving the invariant subspace problem for these spaces. In the field of mathematics known as functional analysis , the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends ...
A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.
These examples shared many common features, and these features were soon abstracted into Hilbert spaces, Banach spaces, and more general topological vector spaces. These were a powerful toolkit for the solution of a wide range of mathematical problems. The most detailed information was carried by a class of spaces called Banach algebras. These ...