Scalar multiplication

Encyclopedia

In mathematics

,

in linear algebra

(or more generally, a module

in abstract algebra

). In an intuitive geometrical context, scalar multiplication of a real

Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. The term "scalar" itself derives from this usage: a scalar is that which scales vectors. Scalar multiplication is different from the scalar product, which is an inner product between two vectors.

from

The result of applying this function to

Here + is addition

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

operation in the field.

Scalar multiplication may be viewed as an external

binary operation

or as an action

of the field on the vector space. A geometric interpretation to

As a special case,

When

The same idea goes through with no change if

and

over

If

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

,

**scalar multiplication**is one of the basic operations defining a vector spaceVector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

in linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

(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, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

in abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

). In an intuitive geometrical context, scalar multiplication of a real

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. The term "scalar" itself derives from this usage: a scalar is that which scales vectors. Scalar multiplication is different from the scalar product, which is an inner product between two vectors.

## Definition

In general, if*K*is a field and*V*is a vector space over*K*, then scalar multiplication is a functionFunction (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

from

*K*×*V*to*V*.The result of applying this function to

*c*in*K*and**in***v**V*is denoted*c***v***.*

Scalar multiplication obeys the following rules(vector in boldface)Scalar multiplication obeys the following rules

*:*+v*Left distributivity*cDistributivityIn mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...

: (*+*d*)*v*=*c*d***v***;*+v*Right distributivity:*c*(*w*+*v*) =*c*c***w***;*);v*Associativity*cdAssociativityIn mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...

: (*)*v*=*c*(*d- Multiplying by 1 does not change a vector: 1
=*v*;*v* - Multiplying by 0 gives the null vector: 0
=*v*;*0* - Multiplying by -1 gives the additive inverseAdditive inverseIn mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....

: (-1)= -*v*.*v*

Here + is addition

Addition

Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . 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....

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 mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....

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 external binary operation 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. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....

or as an action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. 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 transformations 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*K*^{n}, then scalar multiplication is defined component-wise.The same idea goes through with no change if

*K*is a commutative ringCommutative 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 moduleModule (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to 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.