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A graded algebra is an algebra generated when an outer product (wedge product) is defined in a vector space [itex]V_n[/itex] over the scalars [itex]F[/itex].

The outer product generates a set of new entities: the [itex]k[/itex]-vectors. As they are obtained by the outer product of [itex]k[/itex] linearly independent vectors, they are said to be of step or grade [itex]k[/itex]. [itex]k[/itex]-vectors are vectors in nature, so any [itex]k[/itex]-vector is a member of a vector subspace known as subspace of grade [itex]k[/itex], denoted by ∧kVn. Each of this has a dimension of [itex]C(n, k)[/itex] where [itex]C(n, k)[/itex] is the binomial coefficient.

Vectors are said to have step 1, so

[itex]\wedge^1 V_n = V_n,[/itex]
with dimension [itex]n[/itex], and scalars are considered as the 0-step vector space ∧0Vn, and have dimension 1. The [itex]n[/itex]-vectors also generate a 1-dimensional vector space, so all [itex]n[/itex]-vectors are scalar multiples of a arbitrarily-chosen unitary [itex]n[/itex]-vector. Given that essentially behave as scalars, they are often referred to as pseudoscalars. Similarly, [itex](n - 1)[/itex]-vectors are also called pseudovectors.

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:

[itex] \wedge V_n
```  = \wedge^0 V_n + \wedge^1 V_n + \cdots + \wedge^n V_n[/itex]
```
and we call multivectors to its elements.

The dimension of the graded algebra is [itex]2^n[/itex], and the structure of the grades subspaces is that of the Pascal triangle (see binomial coefficient).

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