A confusion with Matrix Multiplication 
Supposed to show how to change basis for matrix.
I traced the arrows with my finger to go from $u_a$ to $v_a$, and I got $T_a=ATA^{-1}$, but that's wrong?
I was also thinking, since transformation is $T_au_a=v_a$, that we apply $A$ then $T$, then $A^{-1}$, so we should have $(A^{-1}(T(Au_a)))$, notice the order of parentheses to show order of transformations.
Is this correct reasoning? Because matrix multiplication is not commutative.
 A: Matrix multiplication is not commutative, but it IS associative, so
$$ (A^{-1}(T(Au_a))) = A^{-1}TAu_a = (A^{-1}TA)u_a $$
When you trace the diagram you find that you have to apply $A$, $T$ and $A^{-1}$, in that order. But, as you've noticed yourself "apply first $P$ then $Q$" corresponds to the matrix product $QP$, so once you know to apply $A$ then $T$ then $A^{-1}$ you need to multiply them together right to left, giving
$$ A^{-1} T A $$
A: If we unwind relevant part of the diagram, we have
$$ \vec{u}_a \stackrel{A}{\longmapsto} \vec{u} \stackrel{T}{\longmapsto} \vec{v} \stackrel{A^{-1}}{\longmapsto} \vec{v}_a $$
(which is meant to be the same as $\vec{u}_a \stackrel{T_a}{\longmapsto} \vec{v}_a $).
This says "take $\vec{u}_a$, apply $A$ to it to get to $\vec{u}$, then apply $T$ to get to $\vec{v}$, then apply $A^{-1}$ to get $\vec{v}_a$." In standard mathematical notation, we read application of functions from right to left, so $f(g(x))$ means "do $g$ to $x$, then do $f$ to the result". So here, we do $A$, then $T$, then $A^{-1}$ to $\vec{u}_a$, i.e.
$$ \vec{v}_a = A^{-1}TA\vec{u}_a. $$
