Final answer:
To find the preimage of a linear transformation, one must identify the known elements, solve the corresponding equations, and verify that the results are reasonable and make sense within the context of the problem.
Step-by-step explanation:
To find the preimage of a linear transformation, begin by outlining the known information from the given problem, such as the transformation equations or the coordinates in the transformed space. Next, solve the appropriate equation or set of equations to find the unknown values, which in this case are the coordinates of the preimage. This can be analogous to solving for a side of a right triangle given the other side and the hypotenuse, where the goal is to 'undo' the function—just like solving for a when a² + b² = c².
For a linear transformation involving two coordinate systems, S and S', as mentioned in Figure 5.13, assume an event is observed in both systems and locate the corresponding coordinates. If necessary, apply the reverse transformation by interchanging the primed and unprimed variables. After solving for the preimage, substitute known values with their units into the equation and calculate the numerical solution, ensuring to use appropriate units such as radians for angles.
Finally, it is essential to verify the reasonableness of the solution. Checking that the answer makes sense in the context of the problem is a critical step to ensure accuracy.