Final answer:
The question involves finding the standard matrix of a linear transformation in mathematics, particularly using projections in a rectangular coordinate system and relating observations in different inertial frames using Lorentz transformations.
Step-by-step explanation:
When trying to find the standard matrix of a linear transformation, it is necessary to determine the effect of the transformation on the basis vectors of the space. A standard matrix represents the linear transformation in terms of how the transformation changes these basis vectors. In this context, we are given vectors in a rectangular coordinate system and would use the equations provided to find the horizontal and vertical components of each vector.
As an example, given a vector à with components Ax, Ay, and Az, the equation to find the components would likely involve projections or dot products, such as X component à · î = Ax, Y component à · ï = Ay and Z component à · â = Az.
For Lorentz transformations, which are used to relate observations in different inertial frames, a specific set of equations, known as the Lorentz transformation equations, is applied to convert the coordinates of an event as observed in one frame to another. The lengths recorded by observers in different frames would also have to consider these transformations.