Tensor Algebra - Non-commutative Polynomials

Non-commutative Polynomials

If V has finite dimension n, another way of looking at the tensor algebra is as the "algebra of polynomials over K in n non-commuting variables". If we take basis vectors for V, those become non-commuting variables (or indeterminants) in T(V), subject to no constraints (beyond associativity, the distributive law and K-linearity).

Note that the algebra of polynomials on V is not, but rather : a (homogeneous) linear function on V is an element of for example coordinates on a vector space are covectors, as they take in a vector and give out a scalar (the given coordinate of the vector).