Final answer:
The question is incomplete, but if it relates to physics and involves solving for the time 't' in a linear transformation, different physics and mathematical concepts may apply, such as kinematic equations or Lorentz transformations in the context of relativity.
Step-by-step explanation:
The question asks us to show that a given linear transformation t is invertible and find a formula for t. However, without specific information about the transformation, such as its definition or properties, it's not possible to give a direct answer. Assuming the transformation pertains to physics, one way to approach invertibility is to demonstrate that the transformation has an inverse function that undoes its actions. If t represents a time-related variable in physics, we may need to apply mathematical methods that are relevant to the physical context, such as kinematic equations or transformations between inertial frames.
For instance, if the transformation was related to motion and we knew the initial position yo and the final position y, we could use kinematic equations to solve for the time t. If the provided quadratic equation t² + 10t - 2000 relates to the transformation, we would use the quadratic formula to determine the value for t.
In cases involving relativistic transformations, such as changing reference frames, the final paragraph hints at using Lorentz transformations to relate the times in different frames, t and t', by the equation t - ux/c² √1-v²/c².