What is the Clifford-Grassmann Geometric Algebra of a Vector Space?

Let us consider a vector space V with dimension n. The outer (or exterior) product of two vectors a and b is defined as the geometric element obtained by parallel transport of the vector a following the way indicated by b. The outer product is indicated by the circumflex accent

a ^ b

If a and b are geometric segments, then a ^ b is the parallelogram formed by both vectors. By repeating this operation of parallel transport we create geometric elements with higher dimension. When we use the components of the vectors in a base of V, the outer product is always a linear combination of the determinants multiplied by the corresponding outer products of the base vectors. For example, in a three-dimensional space, we have:

a = a1e1 + a2 e2 + a3e3         b = b1 e1 + b2e2 + b3 e3

a ^ b = ( a2b3 - a3 b2 ) e2 ^ e3 + ( a3 b1 - a1 b3 ) e3 ^ e1 + ( a1b2 - a2 b1 ) e1 ^ e2

For example if a, b and c are segments, then a ^ b ^ c is the parallelepiped whose three concurrent sides are the three given segments. The outer product is always anticommutative: any permutation of two vectors changes the sign of the outer product (you can read that the permutation of any two rows or columns changes the sign of the determinant). The dimension of the outer products of m vectors is always the combinatorial number Cm, n .
The Grassmann-Clifford Geometric Algebra Cl(V) of a space V is the set of all the elements generated through the outer product of vectors of V, and their linear combinations. From this definition, it is obvious that its dimension is the addition of Cm, n summing over m from zero to n, which yields 2n.
The CGGA has a strong analogy with the Boole's Algebra A(W) of the sample space W, which is the set of all the events formed by joining elementary results of a random experiment. The outer product increases the dimension as soon as the union increases the cardinal of the random event. However, there is another operation whose result has a lower dimension than the factors, the inner product (also called the scalar product). The inner product of vectors is a real number and has the commutative property:

a · b = b · a

whose random analogy is the intersection of events. The geometric product (also called the Clifford product) is defined as:

a b = a · b + a ^ b

and it is represented with no special symbol. The geometric product is associative (also the outer product is associative but not the inner product). The Cl(V) is a closed set with regard to the addition and product of vectors, that is, the result of any sum or product (inner, outer or geometric one) of elements of the geometric algebra is also an element of the geometric algebra. The geometric product is associative and has the unity as the neutral element. Then Cl(V) is an associative algebra with unity element, a type of algebras always having isomorphic matrix representation. Then, the geometric product is isomorphic to the matrix product, a very important result, especially for the programming of computer applications.

You can find more technical explanations on the web What ARE Clifford Algebras and Spinors?