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 2^{n}dimensional vector space equipped with a multiplication operator.