Scalar multiplication

# Scalar multiplication

Discussion

Encyclopedia
In mathematics
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 space
Vector 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 function
Function (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 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 generalizes the distributive law from elementary algebra.For example:...

: (
c + d)
v = cv
+ dv;
• Right distributivity: c(v + w) = cv + cw;
• Associativity
Associativity
In 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...

: (
cd)
v = c(dv
);
• Multiplying by 1 does not change a vector: 1v = v;
• Multiplying by 0 gives the null vector: 0v = 0;
• Multiplying by -1 gives the additive inverse
In 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.

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 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, 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.