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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]).
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.
An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an abelian group under addition, and the four remaining axioms (related to the scalar multiplication) say that this operation defines a ring homomorphism from the ...
In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A.This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K).
[c] Multiplication of quaternions is associative and distributes over vector addition, but with the exception of the scalar subset, it is not commutative. Therefore, the quaternions H {\displaystyle \mathbb {H} } are a non-commutative, associative algebra over the real numbers.
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.
In linear algebra, the closure of a non-empty subset of a vector space (under vector-space operations, that is, addition and scalar multiplication) is the linear span of this subset. It is a vector space by the preceding general result, and it can be proved easily that is the set of linear combinations of elements of the subset.
In other words, any continuous linear functional on is an inner multiplication to some fixed vector, and vice versa. [ citation needed ] Thus, the complex conjugate to a vector v , {\displaystyle v,} particularly in finite dimension case, may be denoted as v † {\displaystyle v^{\dagger }} (v-dagger, a row vector that is the conjugate ...