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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?

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The tensor algebra is generated by formal products $v_1\otimes\cdots\otimes v_n$. If we permute any of the $v_i$, the resulting elementary tensor is not the same. This is a complete lack of commutativity.

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    Apart from minor hazards of the notion of "formal product", one should indeed decide on a definite strategy for proving that $x\otimes y\not=y\otimes x$ for $x,y$ linearly independent. As @Joe Johnson 126's comment indicates, this is the crucial point.2011-11-08