I am just being introduced to tensors (in the context of Cartesian tensors, defined as transforming in a particular way under rotations to a different Cartesian coordinates system). I am reading Riley, Hobson, and Bence (3rd edition) and they state that "it is equally straightforward to show that if the $T_{ij···k}$ are the components of a tensor, then so is the set of quantities formed by interchanging the order of (a pair of) indices, e.g. $T_{ji···k}$". I am trying to make sense of this by supplying a proof but am struggling. I proceed as follows, starting on the left with the transformed components (and where $\textbf{L}$, the transformation matrix, defines the rotation by $x'_i = L_{ij}x_j$ where the $x_i$ are components of a (the) position vector in the space):
$$T'_{ji···k} = (some \, definition \, of \ the \ tensor) = ...$$ However, I'm not sure I even understand what's being defined here. Considering a second-order tensor in two dimensions; surely we're not defining this interchange to mean that $T_{ji}$ is that set of quantities which obtains from assigning $T_{ji} = T_{ij}$ for any $i,j$ (since this would mean defining it as a symmetric tensor)?
