Scalar multiplication

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Scalar multiplication

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, scalar multiplication is one of the basic operations defining a vector space

Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
in linear algebra

Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
(or more generally, a module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalar to lie in a field , the "scalars" may lie in an arbitrary ring....
in abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
). Note that scalar multiplication is different from scalar product which is an inner product between two vectors.

More specifically, if K is a field and V is a vector space over K, then scalar multiplication is a function

Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
from K × V to V. The result of applying this function to c in K and v in V is denoted cv.

Scalar multiplication obeys the following rules (vector in boldface): Here + is addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
either in the field or in the vector space, as appropriate; and 0 is the additive identity in either. Juxtaposition indicates either scalar multiplication or the multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
operation in the field.

Scalar multiplication may be viewed as an external
External (mathematics)

The term external is useful for describing certain algebraic structures. The term comes from the concept of an Binary_operation#External_binary_operations which is a binary operation that draws from some external set....
binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
or as an action
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
of the field on the vector space.

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Encyclopedia

In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, scalar multiplication is one of the basic operations defining a vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
in linear algebra
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
(or more generally, a module
Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalar to lie in a field , the "scalars" may lie in an arbitrary ring....
in abstract algebra
Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
). Note that scalar multiplication is different from scalar product which is an inner product between two vectors.

More specifically, if K is a field and V is a vector space over K, then scalar multiplication is a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
from K × V to V. The result of applying this function to c in K and v in V is denoted cv.

Scalar multiplication obeys the following rules (vector in boldface):
Left distributivity
Distributivity

In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra....
: (c + d)v = cv + dv;
Right distributivity: c(v + w) = cv + cw;
Associativity
Associativity

In mathematics, associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order that the operations are performed does not matter as long as the sequence of the operands is not changed....
: (cd)v = c(dv);
Multiplying by 1 does not change a vector: 1v = v;
Multiplying by 0 gives the null vector
Null vector (vector space)

In linear algebra, the null vector or zero vector is the vector in Euclidean space, all of whose components are zero. It is usually written or 0 or simply 0....
: 0v = 0;
Multiplying by -1 gives the additive inverse
Additive inverse

In mathematics, the additive inverse, or opposite, of a number n is the number that, when addition to n, yields 0 .The additive inverse of F is denoted −F....
: (-1)v = -v.
Here + is addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
either in the field or in the vector space, as appropriate; and 0 is the additive identity in either. Juxtaposition indicates either scalar multiplication or the multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
operation in the field.

Scalar multiplication may be viewed as an external
External (mathematics)

The term external is useful for describing certain algebraic structures. The term comes from the concept of an Binary_operation#External_binary_operations which is a binary operation that draws from some external set....
binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
or as an action
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
of the field on the vector space. A geometric interpretation to scalar multiplication is a stretching or shrinking of a vector.

As a special case, V may be taken to be K itself and scalar multiplication may then be taken to be simply the multiplication in the field. When V is Kⁿ, then scalar multiplication is defined component-wise.

The same idea goes through with no change if K is a commutative ring
Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a Ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
and V is a module
Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalar to lie in a field , the "scalars" may lie in an arbitrary ring....
over K. K can even be a rig, but then there is no additive inverse. If K is not commutative, then the only change is that the order of the multiplication may be reversed from what we've written above.

See also

Statics
Statics

Statics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity....
Mechanics
Mechanics

Mechanics is the branch of physics concerned with the behaviour of physical body when subjected to forces or Displacement , and the subsequent effect of the bodies on their environment....
Product (mathematics)
Product (mathematics)

In the a mathematics, a product is the result of Multiplication, or an expression that identifies divisors to be multiplied. The order in real number or complex number numbers are multiplied has no bearing on the product; this is known as the Commutativity of multiplication....