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Cartesian coordinates identify points of the Euclidean plane with pairs of real numbers. In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted R n or , is the set of all ordered n-tuples of real numbers, that is the set of all sequences of n real numbers, also known as coordinate vectors.
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.
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.
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.
In simple economic models, it is common to assume that there is only a finite number of different commodities, e.g. houses, fruits, cars, etc., so every bundle can be represented by a finite vector, and the consumption set is a vector space with a finite dimension. But in reality, the number of different commodities may be infinite.
Let be a group, and a finite-dimensional vector space over a field (which in classical invariant theory was usually assumed to be the complex numbers). A representation of G {\displaystyle G} in V {\displaystyle V} is a group homomorphism π : G → G L ( V ) {\displaystyle \pi :G\to GL(V)} , which induces a group action of G {\displaystyle G ...
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]
The unit sphere can be replaced with the closed unit ball in the definition. Namely, a normed vector space is uniformly convex if and only if for every < there is some > so that, for any two vectors and in the closed unit ball (i.e. ‖ ‖ and ‖ ‖) with ‖ ‖, one has ‖ + ‖ (note that, given , the corresponding value of could be smaller than the one provided by the original weaker ...