How Does Matrix Multiplication Work?

Another perspective (assuming a linear algebra context): You could also think of it as applying two Linear Transforms in succession (composition):

So it would be the $$\hat{\textbf{\i}}$$ and the $$\hat{\textbf{\j}}$$ of the first transformation after applying applying the second one.

Ah. Thanks for pointing out a 3B1B video I hadn't noticed yet!