Why do we often (as in Rudin's Principles of Mathematical Analysis, p.$~206$) write $A\vec x$ to mean $A(\vec x)$ for linear transformations $A$?

I think it's because we can write $A(\vec x+\vec y)=A\vec x+A\vec y$ and pretend like we have a multiplication that is distributive over addition, which makes thinking about linear transformations easier. My friend insists it's because it simplifies things to identify $A$ with its matrix, and we tend to not write parentheses when multiplying matrices with vectors.