I was just curious in describing the notion that the tensor algebra of a vector space (That is the direct sum of all spaces containing k-tensors for each k) need not be commutative because I am having trouble coming up with an explicit example besides maybe multiplication of matrices.
Let $V\ $ be a finite dimensional vector space such that the dimension of $V\,$ is 2 or greater.
How do you show the tensor algebra $T(V) = \oplus_{k=1}^{\infty} T^k(V)$ is not commutative?