[graycat]

Let's see: Pick a system that we know to be linear. Get a lot of pairs of real inputs and outputs and find the coefficients of the linear function that relates the inputs to the outputs. Then given a new input, can say what the output would be.

Okay, the function, system between a violin on the stage at Carnegie Hall and a seat near the roof is linear. So, I'm typing quickly here, do regression with a recording at the stage and at the seat and look for the coefficients in the convolution. Then given an oboe, can say what it will sound like in the seat.

Hooks law with small deflections is linear. So, take independent variables the forces on a space frame and the dependent variables the deflections and estimate all the spring stiffness values.

Take 200 recipes for tomato sauce all from the same 10 ingredients, for each recipe measure the weight of each ingredient and the weight of the protein in the final sauce and estimate the protein in each of the 10 ingredients. Then for any new tomato sauce recipe, weigh the ingredients and get the protein in the final sauce.