I'm a linear algebra student and I've just come across the formal definition for multiplying matrices as follows:
Let $A = \alpha_{ij}$ be an $l \times m$ matrix over $K$ and let $B = \beta_{ij}$ be an $m \times n$ matrix over $K$. The product $AB$ is an $l \times n$ matrix $C = \gamma_{ij}$ where for $1 \leq i \leq l$ and $1 \leq j \leq n$, $$\gamma_{ij} = \sum_{k=1}^m \alpha_{ik}\beta_{kj} $$
This might sound really silly, but I can't fully understand this definition.
I already know how to multiply any two matrices, but I just can't seem to wrap my head around this definition and how the summation works. When do I input the $i$ and $j$, and in what fashion? Why is the summation up to $m$?
I've tried to use a simplified example to help myself but I still haven't been able to make the connection between how I usually multiply matrices (multiply the first row of $A$ by the columns of $B$ for the first row of entries of matrix $C$, do the same with the next rows of $A$ to obtain the rest of the rows of the matrix) and how it connects to this definition.
Can someone help simplify this definition of matrix multiplication, and really break it down? Thank you.