Recently, I came across the concept of measuring a signal $x \in \mathbb{R}^n$ via a measurement matrix $A \in \mathbb{R}^{m\times n}$, where one has the linear system $y=Ax$ with $y \in \mathbb{R}^m$ as the measured vector. In other words, I have a signal $x$ that I want to sample randomly (e.g., with Gaussian distribution) by applying $A$ and obtaining my $m$ measurements in the vector $y$ (let's say $m<n$). How can I think of this measurement process, as $A$ is random and therefore each measured sample is like a linear combination of $x$ with a row of $A$?
Without the mathematics involved I would just think about a (let's say a sine wave) signal/function where I obtain some function values at different (random) times. But I am confused in the case where I would apply $A$ for the measurement as I would not get the information of one time stamp but rather a combination of all of them. So how is this still related to random sampling (I want to get an intuition), i.e., how would one in practice obtain those measurements?
Any intuition for imaging how this process would look is highly appreciated. Thanks!