[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/product-mathematics-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki21\/product-mathematics-wikipedia\/","headline":"Product (mathematics) – Wikipedia","name":"Product (mathematics) – Wikipedia","description":"before-content-x4 Mathematical form after-content-x4 In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers","datePublished":"2022-09-11","dateModified":"2022-09-11","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1311e6e5de3cc034778e5ba94e7c457c0c51ccd8","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1311e6e5de3cc034778e5ba94e7c457c0c51ccd8","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki21\/product-mathematics-wikipedia\/","wordCount":9552,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Mathematical form       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 30 is the product of 6 and 5 (the result of multiplication), and x\u22c5(2+x){displaystyle xcdot (2+x)} is the product of x{displaystyle x} and        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4(2+x){displaystyle (2+x)} (indicating that the two factors should be multiplied together).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. Matrix multiplication, for example, is non-commutative, and so is multiplication in other algebras in general as well.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.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsProduct of two numbers[edit]Product of a sequence[edit]Commutative rings[edit]Residue classes of integers[edit]Convolution[edit]Polynomial rings[edit]Products in linear algebra[edit]Scalar multiplication[edit]Scalar product[edit]Cross product in 3-dimensional space[edit]Composition of linear mappings[edit]Product of two matrices[edit]Composition of linear functions as matrix product[edit]Tensor product of vector spaces[edit]The class of all objects with a tensor product[edit]Other products in linear algebra[edit]Cartesian product[edit]Empty product[edit]Products over other algebraic structures[edit]Products in category theory[edit]Other products[edit]See also[edit]References[edit]Bibliography[edit]Product of two numbers[edit]The product of two numbers or the multiplication between two numbers can be defined for common special cases: integers, natural numbers, fractions, real numbers, complex numbers, and quaternions.Product of a sequence[edit]The product operator for the product of a sequence is denoted by the capital Greek letter pi \u03a0 (in analogy to the use of the capital Sigma \u03a3 as summation symbol).[1] For example, the expression \u220fi=16i2{displaystyle textstyle prod _{i=1}^{6}i^{2}}is another way of writing 1\u22c54\u22c59\u22c516\u22c525\u22c536{displaystyle 1cdot 4cdot 9cdot 16cdot 25cdot 36}.[2]The product of a sequence consisting of only one number is just that number itself; the product of no factors at all is known as the empty product, and is equal to 1.Commutative rings[edit]Commutative rings have a product operation.Residue classes of integers[edit]Residue classes in the rings Z\/NZ{displaystyle mathbb {Z} \/Nmathbb {Z} } can be added:(a+NZ)+(b+NZ)=a+b+NZ{displaystyle (a+Nmathbb {Z} )+(b+Nmathbb {Z} )=a+b+Nmathbb {Z} }and multiplied:(a+NZ)\u22c5(b+NZ)=a\u22c5b+NZ{displaystyle (a+Nmathbb {Z} )cdot (b+Nmathbb {Z} )=acdot b+Nmathbb {Z} }Convolution[edit]  The convolution of the square wave with itself gives the triangular functionTwo functions from the reals to itself can be multiplied in another way, called the convolution.If\u222b\u2212\u221e\u221e|f(t)|dt\u2212\u221e\u221e|g(t)|dtg)(t):=\u222b\u2212\u221e\u221ef(\u03c4)\u22c5g(t\u2212\u03c4)d\u03c4{displaystyle (f*g)(t);:=int limits _{-infty }^{infty }f(tau )cdot g(t-tau ),mathrm {d} tau }is well defined and is called the convolution.Under the Fourier transform, convolution becomes point-wise function multiplication.Polynomial rings[edit]The product of two polynomials is given by the following:(\u2211i=0naiXi)\u22c5(\u2211j=0mbjXj)=\u2211k=0n+mckXk{displaystyle left(sum _{i=0}^{n}a_{i}X^{i}right)cdot left(sum _{j=0}^{m}b_{j}X^{j}right)=sum _{k=0}^{n+m}c_{k}X^{k}}withck=\u2211i+j=kai\u22c5bj{displaystyle c_{k}=sum _{i+j=k}a_{i}cdot b_{j}}Products in linear algebra[edit]There are many different kinds of products in linear algebra. Some of these have confusingly similar names (outer product, exterior product) with very different meanings, while others have very different names (outer product, tensor product, Kronecker product) and yet convey essentially the same idea. A brief overview of these is given in the following sections.Scalar multiplication[edit]By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map R\u00d7V\u2192V{displaystyle mathbb {R} times Vrightarrow V}.Scalar product[edit]A scalar product is a bi-linear map:\u22c5:V\u00d7V\u2192R{displaystyle cdot :Vtimes Vrightarrow mathbb {R} }with the following conditions, that "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/product-mathematics-wikipedia\/#breadcrumbitem","name":"Product (mathematics) – Wikipedia"}}]}]