# Simple tensor identity: what does it mean to interchange subscripts?

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)?

I see my error now. We are not asserting that $$T_{ji} = T_{ij}$$; rather, we are defining a new tensor (call it $$D_{ij}$$) by $$D_{ij}=T_{ji}$$.