# Proof that the transpose of a tensor is a tensor

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## Homework Statement

Prove that the transpose of a tensor is a tensor.

## Homework Equations

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

## 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.