Product (mathematics)

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.

Integers allow positive and negative numbers. Their product is determined by the product of their positive amounts, combined with the sign derived from the following rule:

Two fractions can be multiplied by multiplying their numerators and denominators:

The geometric meaning is that the magnitudes are multiplied and the arguments are added.

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.

The convolution of the square wave with itself gives the triangular function

Two functions from the reals to itself can be multiplied in another way, called the convolution.

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

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

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 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.