I am going through a chapter in my book on tensors, and it gives a basic understanding of tensors.
The question posed is simple: "Show that the contracted tensor $T_{ijk}V_k$ is a rank-2 tensor."
I followed the basic steps outlined earlier in the chapter, and cross checked with Slader.
I understand the Slader answer uses the fact that the indicies can be arbitrary to do the math, but I am very new to tensors and am still trying to get the basics down. Before I looked up an answer, I got the following:
$$ \begin{align*} T^{'}_{\alpha\beta\gamma} &= a_{\alpha i} a_{\beta j} a_{\gamma k} T_{ijk}\\ V^{'}_\gamma &= a_{\gamma k} V_k\\ T^{'}_{\alpha\beta\gamma} V^{'}_\gamma &= a_{\alpha i} a_{\beta j} a_{\gamma k} T_{ijk}a_{\gamma k} V_k\\ &= a_{\alpha i} a_{\beta j} a_{\gamma k} a_{\gamma k} T_{ijk} V_k\\ &= a_{\alpha i} a_{\beta j} \delta_{\gamma k} T_{ijk} V_k\\ \end{align*} $$
Can someone offer some insight as to what I am missing?
You are almost there.
Summing over the repeated index $k$ and using the properties of the delta function ... $$a_{\alpha i} a_{\beta j} \delta_{\gamma k} T_{ijk} V_k = a_{\alpha i} a_{\beta j} T_{ij\gamma} V_\gamma$$ which is the transformation rule for a second rank tensor