Defining Matrix Multiplication I'm reading Axler's Linear Algebra Done Right. I'm reading the section where he is defining matrix multiplication:

And I'm kind of confused with all the notation. Would someone be willing to help me through it? Any help/hints appreciated. Thank you
 A: Unpacking that notation:
\begin{align}(S\circ T)(u_k) &= S(T(u_k)) = S(\sum_{r=1}^n C_{r,k}v_r)\tag{1}\\
&= \sum_{r=1}^n C_{r,k}S(v_r)\tag{2}\\
&= \sum_{r=1}^n C_{r,k}\sum_{j=1}^m A_{j,r}w_j\tag{3}\\
(S\circ T)(u_k) &= \sum_{j=1}^m\left(\sum_{r=1}^n A_{j,r}C_{r,k}\right)w_j\tag{4}\end{align}
I tweaked a bit there, using functional notation for the linear operators rather than plain "multiplication". The meaning, though, is all the same.
In line 1, we write out $T(u_k)$ in terms of its known matrix. That's really just the definition of the matrix $C$ there.
Moving to line 2, we move $S$ inside the sum by linearity. Since it's a linear map, $S$ of a linear combination of vectors is that linear combination of $S$ of those vectors.
Moving to line 3, we write out the values of $S(v_r)$ in terms of the matrix $A$ of $S$ - the same thing we did in line 1.
Moving to line 4, we switch the order of summation. In the old order, we had to keep all of the terms with $j$ in them in the inner sum, while in the new order it's the terms with $r$ in them that have to be on the inside and the term $w_j$ can be brought to the outer sum.
Now, at the end, we have $(S\circ T)(u_k)$ written as a sum over $j$ of some coefficients times $w_j$. Those coefficients $B_{j,k}=\sum_{r=1}^n A_{j,r}C_{r,k}$ will then be the entries of the matrix of $S\circ T$, and we use this pattern to define matrix multiplication.
