Geometric Algebra
A geometric algebra can be created from a vector space by introducing a multiplication of vectors. This involves creating new elements within the space and incresing its dimension. If e1 and e2 are original basis vectors, then a new element e12 is created which is define to be the product of e1 and e2 (in that order) and minus the product of e2 and e1 (in the other order). Here e12 is a basis element of the larger space and is called a bivector. If e1, e2 and e3 are original basis vectors, then e123 is defined as their product and is this is a trivector.
More generally, suppose that original vector space has dimension n and that e1, e2, ..., en are are set of basis vectors. Then there is a basis element eS of the geometric algebra for any subset S of the full set of n subscripts. The basis element associated with the empty subset behaves like the real number 1 and is identified with it.
If the original space has dimension n, then the dimension of the geometric algebra is 2 to the power n.
To complete the definition of the multiplication, the squares of the original basis vectors need to be defined. These squares are (usually) taken to be a real number and often +1 or -1 are used.