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 V_n and V_m of n and m dimensions, respectively,

The dual vector space of V_n is another n-space.
The cartesian product of V_n and V_m is an (n+m)-space.
The tensor product of V_n and V_m is an (nm)-space.
The Grassmann algebra for V_n is a 2ⁿ dimensional vector space equipped with a multiplication operator.

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.