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Scalar multiplication of a vector by a factor of 3 stretches the vector out. The scalar multiplications −a and 2a of a vector a. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra [1] [2] [3] (or more generally, a module in abstract algebra [4] [5]).
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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 members of R are sometimes called scalars, and the binary operation of scalar multiplication is a function R × M → M, which satisfies several axioms. Counting the ring operations these systems have at least three operations. Vector space: a module where the ring R is a field or, in some contexts, a division ring.
A quaternion of the form a + 0 i + 0 j + 0 k, where a is a real number, is called scalar, and a quaternion of the form 0 + b i + c j + d k, where b, c, and d are real numbers, and at least one of b, c, or d is nonzero, is called a vector quaternion. If a + b i + c j + d k is any quaternion, then a is called its scalar part and b i + c j + d k ...
In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective space P(V), whose elements are one-dimensional subspaces of V.More generally, any subset S of V closed under scalar multiplication defines a subset of P(V) formed by the lines contained in S and is called the projectivization of S.
The operation · is called scalar multiplication. Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in R. One may write R M to emphasize that M is a left R-module. A right R-module M R is defined similarly in terms of an operation · : M × R → M.
Given a curve, E, defined by some equation in a finite field (such as E: y 2 = x 3 + ax + b), point multiplication is defined as the repeated addition of a point along that curve. Denote as nP = P + P + P + … + P for some scalar (integer) n and a point P = (x, y) that lies on the curve, E. This type of curve is known as a Weierstrass curve.