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A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
The space of vectors may be considered a coordinate space where elements are associated with a list of elements from K. The units of the field form a group K × and the scalar-vector multiplication is a group action on the coordinate space by K ×. The zero of the field acts on the coordinate space to collapse it to the zero vector.
The Outer space, denoted X n or CV n, comes equipped with a natural action of the group of outer automorphisms Out(F n) of F n. The Outer space was introduced in a 1986 paper [ 1 ] of Marc Culler and Karen Vogtmann , and it serves as a free group analog of the Teichmüller space of a hyperbolic surface.
There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V. Inverse elements of vector addition: For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0. Compatibility of scalar multiplication with field multiplication: a(bv) = (ab)v [nb 3]
In mathematics, the exterior algebra or Grassmann algebra of a vector space is an associative algebra that contains , which has a product, called exterior product or wedge product and denoted with , such that = for every vector in .
An (R,S)-bimodule is an abelian group together with both a left scalar multiplication · by elements of R and a right scalar multiplication ∗ by elements of S, making it simultaneously a left R-module and a right S-module, satisfying the additional condition (r · x) ∗ s = r ⋅ (x ∗ s) for all r in R, x in M, and s in S.
The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space.
However, in the special case of a conservative vector field, the value of the integral is independent of the path taken, which can be thought of as a large-scale cancellation of all elements that do not have a component along the straight line between the two points. To visualize this, imagine two people climbing a cliff; one decides to scale ...