There are several mathematical processes to derive a vector space from others. The new spaces are over the same field. All of these constructions are independent of coordinate systems, but sometimes it is convenient to use a coordinate system to define or specify the construction. Given vector spaces Vn and Vm of n and m dimensions, respectively,

The Clifford construction starts with an n-space V and a quadratic form for that space and arrives at a 2n-space C with a multiplication operator. This is a vector space with a multiplication, which is called an algebra. There are several ways to get from V to C.