I am reading Sheldon Axler's "Linear Algebra Done Right, Third Edition". I am looking at the page where he defines matrix multiplication. He shows the motivation behind the way it is defined, as a composition of Linear Maps. I intuitively understand it; to show the effect U1 has on W1 you use T to impress U1 on the basis vectors of V, and then you use S to impress the combination of the basis vectors of V onto W1. So I have no problem actually performing the matrix multiplication, and I have no problem understanding it intuitively. But I have one issue understanding his derivation of it.
This is the excerpt from the book
How does he get from the third line to the fourth line? I don't understand the rules of shifting around summation symbols. he shifted the order of the scalars and of the summation symbols.
\begin{align} \sum_{r=1}^{n}C_{r,k}\left(\sum_{j=1}^{m}A_{j,r}w_j\right)&=\nonumber\\ =&\sum_{r=1}^{n}C_{r,k}\left(A_{1,r}w_1+\cdots+A_{m,r}w_m\right)\nonumber \\ =&C_{1,k}\left(A_{1,1}w_1+\cdots+A_{m,1}w_m\right)+\cdots+C_{n,k}\left(A_{1,n}w_1+\cdots+A_{m,n}w_m\right)\nonumber\\ =&\left(A_{1,1}C_{1,k}w_1+\cdots+A_{m,1}C_{1,k}w_m\right)+\cdots+\left(A_{1,n}C_{n,k}w_1+\cdots+A_{m,n}C_{n,k}w_m\right)\nonumber\\ =&\left(A_{1,1}C_{1,k}w_1+\cdots+A_{1,n}C_{n,k}w_1\right)+\cdots+\left(A_{m,1}C_{1,k}w_m+\cdots+A_{m,n}C_{n,k}w_m\right)\nonumber\\ =&\left(A_{1,1}C_{1,k}+\cdots+A_{1,n}C_{n,k}\right)w_1+\cdots+\left(A_{m,1}C_{1,k}+\cdots+A_{m,n}C_{n,k}\right)w_m\nonumber\\ =&\left(\sum_{r=1}^{n}A_{1,r}C_{r,k}\right)w_1+\cdots+\left(\sum_{r=1}^{n}A_{m,r}C_{r,k}\right)w_m\nonumber\\ =&\sum_{j=1}^{m}\left(\sum_{r=1}^{n}A_{j,r}C_{r,k}\right)w_j \end{align}