The outer product generates a set of new entities: the -vectors. As they are obtained by the outer product of linearly independent vectors, they are said to be of **step** or **grade** . -vectors are vectors in nature, so any -vector is a member of a vector subspace known as subspace of grade , denoted by ∧^{k}*V _{n}*. Each of this has a dimension of where is the binomial coefficient.

Vectors are said to have step 1, so

In order to achieve closure, all these spaces are combined by considering the direct sum of all of them. The resulting space is a new vector space called the **graded algebra**:

The dimension of the graded algebra is , and the structure of the grades subspaces is that of the Pascal triangle (see binomial coefficient).