# Proof that the transpose of a tensor is a tensor

1. Nov 10, 2011

### wonfish

11. The problem statement, all variables and given/known data
Prove that the transpose of a tensor is a tensor.

2. Relevant equations
Definition of the transpose: a$\bullet$Tb = b$\bullet$T^Ta where a and be are arbitrary vectors

3. The attempt at a solution
This isn't homework per se, I'm 60 yo and studing continuum mechanics, but I see that I can't use any properties of transposes since I haven't proven first that T^T is a tensor. Likewise I can't use properties of tensors either except for the linear property defintion of a tensor : T(αa + βb) = αTa + βTb

This definition doesn't seem to get me anywhere.