Product (mathematics)

In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors.

For example, 21 is the product of 3 and 7 (the result of multiplication), and

When one factor is an integer, the product is called a multiple.

The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication.

When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors.

There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matrices, one can also define products on many different algebraic structures.

Originally, a product was and is still the result of the multiplication of two or more numbers.

The fundamental theorem of arithmetic states that every composite number is a product of prime numbers, that is unique up to the order of the factors.

With the introduction of mathematical notation and variables at the end of the 15th century, it became common to consider the multiplication of numbers that are either unspecified (coefficients and parameters), or to be found (unknowns).

These multiplications that cannot be effectively performed are called products.

Later and essentially from the 19th century on, new binary operations have been introduced, which do not involve numbers at all, and have been called products; for example, the dot product.

The product operator for the product of a sequence is denoted by the capital Greek letter pi Π (in analogy to the use of the capital Sigma Σ as summation symbol).

Under the Fourier transform, convolution becomes point-wise function multiplication.

A scalar product is a bi-linear map: with the following conditions, that

From the scalar product, one can define a norm by letting

The scalar product also allows one to define an angle between two vectors: In

-dimensional Euclidean space, the standard scalar product (called the dot product) is given by: The cross product of two vectors in 3-dimensions is a vector perpendicular to the two factors, with length equal to the area of the parallelogram spanned by the two factors.

The cross product can also be expressed as the formal[a] determinant: A linear mapping can be defined as a function f between two vector spaces V and W with underlying field F, satisfying[3] If one only considers finite dimensional vector spaces, then in which bV and bW denote the bases of V and W, and vi denotes the component of v on bVi, and Einstein summation convention is applied.

Now we consider the composition of two linear mappings between finite dimensional vector spaces.

Then one can get Or in matrix form: in which the i-row, j-column element of F, denoted by Fij, is fji, and Gij=gji.

The composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication.

In other words: the matrix product is the description in coordinates of the composition of linear functions.

Given two finite dimensional vector spaces V and W, the tensor product of them can be defined as a (2,0)-tensor satisfying: where V* and W* denote the dual spaces of V and W.[4] For infinite-dimensional vector spaces, one also has the: The tensor product, outer product and Kronecker product all convey the same general idea.

In general, whenever one has two mathematical objects that can be combined in a way that behaves like a linear algebra tensor product, then this can be most generally understood as the internal product of a monoidal category.

That is, the monoidal category captures precisely the meaning of a tensor product; it captures exactly the notion of why it is that tensor products behave the way they do.

More precisely, a monoidal category is the class of all things (of a given type) that have a tensor product.

The empty product on numbers and most algebraic structures has the value of 1 (the identity element of multiplication), just like the empty sum has the value of 0 (the identity element of addition).

However, the concept of the empty product is more general, and requires special treatment in logic, set theory, computer programming and category theory.

For the general treatment of the concept of a product, see product (category theory), which describes how to combine two objects of some kind to create an object, possibly of a different kind.