Tensors applied to vector and dual vector fields in GR This is a specific question about tensor manipulation in Wald's GR. I'm asking for clarification of a trivial step, because I'm working through the text outside the context of a class, without prior background in diff. geometry. 
On pg33 in Wald, going from eq.3.1.11 to eq.3.1.12, somehow $(C^c_{ab}\omega_c)t^b$ becomes $\omega_b (C^b_{ac}t^c)$ from index substitution on contracted indices. I see that the switch $b \leftrightarrow c $ was made, but what justifies bringing $\omega_c$ to the left of $C^c_{ab}$? I know that $(C^c_{ab}\omega_c)t^b$ evaluates to a type (0,1) tensor, but what is $\omega_b (C^b_{ac}t^c)$?
The only guess that makes sense to me is that $\omega_b C^b_{ac} = C^b_{ac} \omega_b$. Can anyone confirm? Thank you.
 A: Yes, you're exactly right, $\omega_b {C^b}_{ac} = {C^b}_{ac} \omega_b$.  In general, the order of factors doesn't matter in a tensor expression like this.
This is Wald, so technically you're supposed to think of those expressions as using abstract index notation instead of involving components in any particular basis, but it's also valid to think of them as involving components, and it's arguably easier to understand the equality if you do so.  It's perhaps also easier to see the equality if you consider any arbitrary choice of indices for $a$ and $c$.  For example, with the arbitrary choice of $a=1$ and $c=2$, all $\omega_b {C^b}_{ac} = {C^b}_{ac} \omega_b$ is saying is that
$$\sum_{b}\omega_b {C^b}_{12}=\sum_{b} {C^b}_{12} \omega_b\ ,$$
where $\omega_0 ... \omega_3$ and ${C^0}_{12} ... {C^3}_{12}$ are just two sequences of four real constants.  All that equation is really expressing is just the commutivity of reals under multiplication.
The choice of which indices you looked at in the above was arbitrary, so the equality is valid if you express it generically in terms of arbitrary indices.  And the choice of basis used in order to think of the equation as involving components was arbitrary, so the equation remains valid if you think of it as involving abstract index notation in which the basis isn't specified.
To answer your other question, $\omega_b (C^b_{ac}t^c)$ is also a type $(0,1)$ tensor.
