Basis of Clifford algebra

Question

As picture below, I know the $T(V)=\bigoplus_{_{k\ge 0}} V\otimes ...\otimes V$ is a vector space,

and $I=\{T_1\otimes(v\otimes v +||v||^2)\otimes T_2 , v\in V , T_1,T_2\in T(V)\}$ is the ideal ,

so $I$ is a subspace. So $T(V)/I$ is a vector space .

Besides, I know the basis of $T(V)$ has the form of red line . But how to kow the basis of Cl(v) has the form of red line ?

Answer 1

We know basis of $T(V)$ is the set of "non-commutative monomials" $e_{i_1} \otimes \dots \otimes e_{i_k}$. By using relations in Clifford algebra you can observe that

1. It is always possible to rearrange vectors (use 2.4.1 or $e_i e_j =-e_j e_i$) in these monomials and get increasing subscripts, possibly changing the sign.
2. If you get two repeated vectors in monomial, applying Clifford relations give you a monomial of degree $k-2$ (use $e_i^2=-1$).

So, you end up with monomials of the form given in the book and this monomials are linearly independant.